1 Insee Références -2023 Edition -Dossiers -Religious diversity in France: intergenerational transmissions and practices by origins

Religious diversity in France: intergenerational transmissions and practices by origins

In 2019-2020, 51% of the population aged 18 to 59 in metropolitan France said they had no religion. This religious disaffiliation has been increasing over the past ten years and concerns 58% of people with no migration background, 19% of immigrants who arrived after the age of 16 and 26% of the descendants of two immigrant parents.

While Catholicism remains the dominant religion (29% of the population declare themselves to be Catholic), Islam confirms its place as the second religion in France (10%). The number of people claiming another Christian religion is also increasing, reaching 9%. The frequency and intensity of religious practice varies by religious affiliation: only 8% of Catholics regularly attend a place of worship, compared to just over 20% of other Christians, Muslims and Buddhists, and 34% of Jews.

Processes of religious transmission between generations shape the religious landscape over the long term: 91% of people raised in a Muslim family follow the religion of their parents. This transmission is also very strong among Jews (84%), but less so among Catholics (67%) and other Christians (69%). All else being equal, growing up in a family of mixed religious or Catholic ancestry is a key factor in the secularization process among the descendants of immigrants.

The place of religions in France has considerably evolved since the end of the 1950s, following a double process of secularization (a reduction of the role of religions in the organization of social life and a decrease in the number of people who consider themselves affiliated with a religion) and religious diversification. The statistical landscape of religions in France is not well known, as this information is not collected in the Census and rarely appears in official statistics insert 1. The Trajectoires et Origines survey is one of the rare data sources that is representative of the population residing in metropolitan France and that allows for the study of religious affiliations and practices source. By covering different origins and immigrant generations, it makes it possible to study how immigrant families, families of immigrant descent, as well as the rest of the population, relate to religion and its transmission over generations.

Immigrants are twice as likely to affiliate with a religion compared to those without a migration background

In 2019-2020, 29 percent of people aged 18-59 identify as Catholic, 10 percent as Muslim, and 10 percent as having other religious affiliations, while the remaining 51 percent identify as having no religion. This distance from religion varies considerably depending on the link to migration: 58% of those with no migration background over two generations say they have no religion, compared to only 19% of immigrants who arrived after the age of 16 and 26% of descendants of two immigrant parents figure 1. This general observation must be qualified according to origin: immigrants from countries with a Muslim tradition (Maghreb, Turkey, Sahelian Africa) are the most religiously affiliated, while those from predominantly Christian European countries or from Asia (especially China) are less likely to declare a religious affiliation figure 2. Among the predominantly Christian countries, immigrants from Portugal and Central Africa are exceptions, with more religious affiliation (79% and 77% Christian, respectively). While Catholicism remains the dominant religion among Christian denominations in France, Protestant churches have been growing in Central Africa and are more present in many European countries; other Christian religions are thus overrepresented among immigrants from these regions. They are also gaining ground among people from French overseas territories, with 18% of overseas natives (38% Catholic) and 18% of their descendants residing in metropolitan France

Insert 1 -Measuring Religion and Religious Practices in Quantitative Surveys

Religious affiliation and religious practices are not generally measured in official statistical sources. Although data relating to religion are considered "sensitive" under the terms of law no. 78-17 of January 6, 1978 on information technology, files and freedoms, their processing is authorized "for scientific or historical research or statistical purposes", for purposes determined and made explicit by the CNIL (Commission Nationale de l'Informatique et des Libertés), and if the data collected is adequate, relevant and not excessive in relation to its purposes.

A question on religion appeared in the census until 1872 and was abandoned thereafter (except in Alsace concordataire -outside the purview of secularism law -until 1962 1 ). However, some surveys collected information on religion, such as the ERFI survey (Survey on Family and Intergenerational Relations, INED, French component of the European Gender and Generation survey, [2005][2006][2007][2008][2009][2010][2011] or the EPIC survey (Study of individual and marital paths, INSEE-Ined, 2013-2014). Questions on religion are often present in election studies relying on surveys produced by various political science laboratories in France, but also in other European countries and in the United States, since religion is one of the constituent variables of the sociological models of voting developed in the 1960s and 1970s. They are also collected in several European surveys such as the European Social Surveys (ESS) and the European Value Studies, the latter dating back to the early 1990s.

In the Trajectoires et Origines survey (TeO), religion is declared by respondents in response to the question "Do you have a religion today?", and another open-ended question (without a list of religions provided) asks which one. This wording follows that used in the ESS. Several questions then describe religious practices and their frequency, making it possible to measure the religiosity of the respondent and that of the family in which he or she grew up. Although responses to the survey are mandatory, interviewers inform respondents that responses to the religion section are optional: 1% of respondents did not wish to answer the question "Do you have a religion?". The inclusion of questions on religion has been validated by the CNIL.

1 For a review of statistical sources on religion in the late 1980s, see [START_REF] Djider | La vie religieuse : chiffres et enquêtes[END_REF]. (33% Catholic). Muslims now represent 10% of the population in metropolitan France, but are much more present among immigrants (44%) and the descendants of two immigrant parents (48%). 14% of the descendants of immigrants from mixed couples and 1% of the population without migratory ancestry claim a Muslim affiliation.

Persons reporting no religion by migration background in

More people without a religion since 2008

A comparison of the 2019-2020 results with those of the first TeO survey in 2008-2009 highlights how the secularization trend has continued [START_REF] Simon | Sécularisation ou regain religieux : la religiosité des immigrés et de leurs descendants[END_REF]. In the age range common to both surveys, the 18-49 year-olds, the proportion of people who declare themselves to be without religion has increased from 45% to 53% in 10 years figure 1. This increase varies according to the migration background: it is very pronounced among people with no migratory ancestry (+12 points), but is only of 3 points for immigrants who arrived in France as adults. For the descendants of two immigrant parents, this share has not changed. These contrasting trends reflect very heterogeneous secularization dynamics across origin groupes and religious affiliations.

Compared to 2008-2009, a relatively distanced relationship to religion can be measured through the place it occupies in people's identity. When respondents are asked to name up to four dimensions of their identity (gender, social class, level of education, family situation, nationality, etc.), in 2019-2020, 7% of people aged 18-49 cite religion figure 3, the same as in 2008-2009. This place of religion in identity is clearly greater for Jews (54%) and Muslims (30%) than for Catholics (6%), but it has declined in eleven years among Muslims.

Religious affiliation by migration background and origin

Choice of religion as a dimension of identity, by religious affiliation, in 2008-2009 and 2019-2020

in %

2008-2009 2019-2020

Religious affiliation

Religion as a dimension of identity

Religious affiliation

Religion as a dimension of identity Catholics are less consistent in their religious practices, while Muslims are more active.

The study of religious practices in France has long focused on Catholicism, the dominant religion, and religious attendance has been important for distinguishing regular from occasional churchgoers, non-practitioners, and even "cultural Catholics," who express a family tradition rather than a religious attachment per se.

In 2019-2020, 8% of Catholics regularly attend their place of worship figure 5a. This is the case for just over 20% of other Christians, Muslims and Buddhists, and 34% of Jews. Within the same religion, practices differ according to migration background: 6% of Catholics with no migration background go to mass regularly, compared to 15% of immigrants of Portuguese, Spanish or Italian origin (the rate is only 5% among their descendants), 24% of Catholics from overseas who have migrated to metropolitan France, and up to 55% of immigrants from Central Africa. Attending the mass is more widespread among non-Catholic Christians: 62% of Central African immigrants are regular churchgoers.

Places of worship have a different role in Islam, where private practices, at home or in daily life, are much more frequent. Attending the mosque is not one of the five pillars of Islam, and only 20% of Muslims attend it regularly. Variations by migration background and origins are rather small.

Religious practices by religion a. Attendance at places of worship

Regular

For religious holidays For family ceremonies Never The dissociation between religious attendance and the declared importance of religion in one's life shows that the former is a poor measure of Muslims' religiosity. Thus, 76 percent of Muslims say that religion is very or somewhat important to them, compared to 27 percent of Catholics and 39 percent of other Christians. However, while 47 percent of Catholics and 76 percent of other Christians for whom religion is very important regularly attend a place of worship, only 34 percent of Muslims who consider religion very important do so. Among Muslims overall, 19% combine the importance of religion with regular attendance at places of worship, while 57% consider religion to be important in their lives but never go to the mosque, or only for holidays or special ceremonies.

Conversely, Catholics who attend church and consider religion important are 7% of all Catholics while those who consider religion important without attending religious service are only 20%1 .

For Muslims, the dissociation between mosque attendance and religiosity is strongly linked to gender: 30% of Muslim men and 10% of Muslim women regularly attend the mosque, while the latter declare slightly more often (78%) than the former (73%) that religion is important in their lives. Among Catholics, 8% of men and 9% of women attend mass regularly.

Insert 2 -Who wears the veil?

The wearing of the veil by Muslim women is an important aspect of public debates on the visibility of Islam and gender equality. The TeO survey provides a measure of this practice: 26% of Muslim women between the ages of 18 and 49 say they wear a headscarf figure. Among them, nine out of ten say they always wear it. This practice varies greatly between immigrants of one origin and descendants of that same origin; it concerns 36% of Muslim immigrant women and 17% of second-generation descendants. The practice is most common among Turkish and Middle Eastern immigrants (46%). While veiling is more common among immigrant women after the age of 35 (30% among 18-24 year olds and 42% among 35-44 year olds), it is among 25-34 year olds that veiling is most prevalent among the descendants of immigrant (20%, compared to 17% among 35-44 year olds). Compared to 2008-2009, the wearing of the veil has become more frequent for all origins and all generations. These increases primarily reflect changes in practices in the countries of origin, which is visible for immigrant women, and a smaller increase for second generation women. Muslim women in couples are wearing the veil more often. This is the case for 42% of immigrant women of North African origin in couples, compared to 24% of those living alone. Wearing the veil is more common among inactive Muslim women: 56% among immigrants and 39% among descendants of immigrants. Finally, the wearing of the veil differs greatly by professional category: from 13% for Muslim women in management and intermediate professions to 38% for working-class women. Insee Références -2023 Edition -Dossiers -Religious diversity in France: intergenerational transmissions and practices by origins

Wearing of the veil among Muslim women, by migration background and origin

Religiosity outside of religious institutions manifests itself in the practice of prayer and the observance of various obligations, including dietary and clothing obligations insert 2. Muslims pray most frequently: 58% at least once a week, while only 21% never pray figure 5b. Buddhists have the fewest believers who never pray (15%), but only 32% pray at least once a week. Finally, prayer is much less common among those who identify as Catholic: 44% never pray and only 15% pray at least once a week. In Catholicism and other Christian denominations, the attendance of church service and the practice of prayer are closely linked: among Catholics, 64% of those who go to church regularly pray at least once a week, compared to 5% of those who do not go to mass. This correlation is weaker in the Muslim religion, for which the observance of prayer is a requirement: while 58% of Muslims pray at least once a week, this is still the case for 35% of those who never go to the mosque. Not all religions advocate dietary restrictions, but fasting periods exist in most religions (the Lent fast, Ramadan, Yom Kippur or Pesach, the invitation to eat lean on Fridays). Fasting practices have practically fallen into disuse in Christian denominations: only 3% of Catholics and 5% of other Christians say that they strictly respect the fasting periods of their religion and 13% respect them "more or less". The few Christians who still practice fasting are immigrants from Central Africa (in this group, 20% practice it strictly). In contrast, 75% of Muslims observe the fast strictly and 15% "more or less".

A stronger transmission in Muslim and Jewish families

Immigrant arrivals contribute to increasing religious diversity in France. Over the long run however, the religious landscape is primarily shaped by processes of religious transmission between generations. Transmission dynamics depend on the motivation of immigrant parents to socialize their children in their religion. Meanwhile, incentives for secularization are relatively strong in France.

Religious reproduction within the family is strong in Islam and Judaism: 91% of those raised in Muslim families and 84% in Jewish families continue to identify with their parents' religion figure 6. Christian families transmit their religion at a lower rate: 67% of those raised by Catholic parents and 69% by parents of other Christian denominations retained their parents' religion. These transmission rates by religion also vary by origin group. In Catholic families from overseas territories, transmission rates are higher (71% for the second generation) than that observed in families with no migratory ancestry (66%). Similarly, Muslim families from the Maghreb transmit their religion slightly less (89%) than those from Turkey/the Middle East or Sahelian Africa (97%).

Religious affiliation by parents' religion

The majority of those who did not follow their parents' affiliation left religion altogether rather than adopt another. The "exit rate" from religion is particularly high (57%) among those whose parents have two different religions or those in which one parent has no religion. Such parental religious mixing occurs among 16% of both the second generation and the population without a migration background. It amounts to 14% of second-generation Catholics and only 6% of second-generation Muslims.

Religiosity mainly depends on parental religious socialization

Like religious affiliation, religiosity is strongly influenced by family socialization. 60% of those with no religion report having received a parental upbringing in which religion was not important figure 7. More than half of Catholics, other Christians, and Buddhists responded that it was of little or no importance in their childhood. Parental religious socialization is significantly more consistent for Muslims and Jews, as nearly three-quarters of them say it was somewhat or very important. To provide a better understanding of the link between religious socialization and religiosity in adulthood among the descendants of immigrants born and raised in France, we adopt a "all other things being equal" statistical approach. As in the case of the transmission of affiliations, Muslim religious socialization stands out through the strength of its reproduction [START_REF] Drouhot | Cracks in the Melting Pot? Religiosity and Assimilation Among the Diverse Muslim Population in France[END_REF]: the descendants of Muslim immigrants who grew up in families with strong religious socialization (42% of Muslim families) have a high probability (70%) of declaring that religion has a very important place in their lives figure 8. This remains true when differentiating by gender. In contrast, descendants of immigrants who grew up in Catholic families with strong religious socialization (13% of Catholic families) have only a 21% chance of reporting that religion is very important in their lives. Regardless of the parent's religious affiliation, respondents with mixed religious ancestry show a low likelihood of declaring religion to be important (27%). Parental religious mixedness is thus a powerful secularization force.

Importance of religion in the family during childhood, by respondent's religion

4.

  Proportion of individuals who cited religion as a significant part of their identity, by age cohort in 2008-2009 and 2019-2020 and by religion age cohorts are indexed to 2019-2020. Therefore, subtract 11 years for age in 2008-09 (the 28-34 age cohort in 2019-2020 was 17-23 in 2008-09). Interpretation: 34% of Muslims aged 24-33 in 2008-2009 cited religion as a dimension of their identity. Eleven years later (in 2019-2020), 28% of Muslims aged 35-44 cited it as a dimension of their identity. Scope: Metropolitan France, persons aged 18 to 59 living in ordinary housing. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) and Trajectoires et Origines (2008-2009) surveys.

  Among people who declared themselves to be Catholic, 8% attend church regularly (at least once a month) and 15% pray at least once a week. Scope: Metropolitan France, persons aged 18 to 59 living in ordinary housing and having declared a religion. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) survey.

  The "Mixed" category corresponds to cases where the parents are of two different religions or with a parent of no religion. Single-parent families receive the religion of the parent in the family. Interpretation: 31% of people raised in a Catholic family declare that they have no religion and 67% consider themselves Catholic. Scope: Metropolitan France, persons aged 18 to 59 living in ordinary housing. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) survey.

  60% of the persons declaring to have no religion were raised in a family where religion was not important at all. Scope: Metropolitan France, persons aged 18 to 59 living in ordinary housing. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) survey.

  weighted numbers (n = 5639 observations). Vertical bars represent 95% confidence intervals. Mixed ancestry refers to cases where parents are of two different religions or with one parent of no religion. Predicted probabilities are from a logistic regression model controlling for age and sex. Variables other than religious ancestry and the importance of religion in the education received are fixed at their mean value. Interpretation: All other things being equal, descendants of immigrants who grew up in Muslim families where religion was very important have a 70% probability to report that religion is very important in their lives. Scope: Metropolitan France, descendants of immigrants aged 18 to 59 living in ordinary housing and from families with two parents who are either Catholic, Muslim or of mixed religion. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) survey.

2008-2009 and 2019-2020 in % Age 18 to 49 Age 18 to 59

		2008-2009	2019-2020	2019-2020
	Immigrants who arrived in France after 16	16	19	19
	Immigrants who arrived in France at age 16 or before	24	27	27
	Descendants of two immigrant parents	23	23	26
	Descendants of one immigrant parent	48	54	54
	Neither immigrants nor descendants of immigrants	50	62	58
	Total	45	53	51

Interpretation: 16% of immigrants aged 18 to 49 who arrived in France after 16 years of age declared themselves to be without religion in 2008-2009. They are 19% in 2019-2020. Scope: Metropolitan France, persons aged 18 to 49 or 18 to 59 living in ordinary housing. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) and Trajectoires et Origines (2008-2009) surveys.

French overseas territories Algeria Morrocco or Tunisia Sub-Saharan Africa: Sahel Sub-Saharan Africa: Guinean region or Central Africa Other African countries Southeast Asia Turkey or the Middle-East China 1 Other Asian countries Portugal Spain or Italy Other EU-27 country Total

  Numbers to isolate the descendants of immigrants from China are insufficient.

	No religion	Catholics	Other Christians	Muslims		Jews	Buddhists		Other religions
		Wihout migration background
			Natives
			Descendants of natives
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Immigrants
		Descendants of immigrants
			Total
			0	10	20	30	40	50	60	70	80	90	100
												in %
	1

Interpretation: 33% of overseas natives declare that they have no religion, 38% say they are Catholic and 18% have another Christian affiliation. Scope: Metropolitan France, persons aged 18 to 59 living in ordinary housing. Source: Ined-Insee, Trajectoires et Origines 2 (2019-2020) survey.

  Metropolitan France, persons aged 18 to 49 living in ordinary housing. Sources: Ined-Insee, Trajectoires et Origines 2 (2019-2020) and Trajectoires et Origines (2008-2009) surveys.Such a variation in religious identity can also be measured by pseudo cohorts between2008-2009 and 2019-2020, that is, by reconstructing the age group of 2008-2009 respondents by substracting 11 years from those in 2019-2020. This allows us to distinguish between age effects (the relationship to religion varies according to age, and thus the evolution of the age composition of the population could explain the variation in religious identity) and generation effects (the differences between age groups could actually express specific behaviors of generational groups linked to their religious socialization). The decline in the declaration of religion in the identity of Muslims occurs regardless of the cohort being considered: the stronger religious identity observed eleven years ago among those under 30 years of age compared to those over 30 years of age did not amount to a generation effect bound to continue. Rather, it was the result of an age effect observed again in 2019-2020 figure 4. Religious identity thus decreases with age among Muslims, while it is stable for Catholics.

	No religion	45	1	53	1
	Catholics	43	8	25	6
	Other Christians	2,5	10	9	16
	Muslims	8	33	11	30
	Jews	0,5	46	0,5	54
	Buddhists	0,5	20	0,5	26
	Total	100	7	100	7
	Interpretation: in 2019-2020, 6% of people declaring themselves to be Catholic cite religion as a constitutive dimension of their
	identity.
	Scope:

  Metropolitan France, women aged 18 to 49 living in ordinary housing who declared they were Muslims. Sources: Ined-Insee, Trajectoires and Origines 2 (2019-2020) and Trajectoires etOrigines (2008Origines ( -2009) ) surveys.

				in %
			2008-2009	2019-2020
	Maghreb	Immigrants Descendants of immigrants	23 12	37 16
	Sub-Saharan Africa	Immigrants Descendants of immigrants	12 11	28 17
	Turkey and the Middle East	Immigrants Descendants of immigrants	39 18	46 24
	All origins	Immigrants Descendants of immigrants	22 13	36 17
	Total		18	26
	Interpretation: in 2008-2009, among immigrant women from the Maghreb who declared themselves to be Muslim,
	23% declared that they wore the veil.
	Scope:

Among those who say religion is important in their lives,[START_REF] Tiberj | Chapitre 1. The Muslims next door. Portraits d'une minorité religieuse française[END_REF] distinguishes "private Muslims" from "mosque Muslims": the former are no less religious than the latter, but are distinctive in their lower or non-existent attendance at places of worship. This distinction can be applied by extension to "private Catholics" and "church Catholics."

Authors:

Lucas Drouhot (Utrecht University) Patrick Simon (Ined et ICM) Vincent Tiberj (Sciences Po Bordeaux)

Definition

The affiliation transmission rate is the share of people who retained their parents' religious affiliation.

Source

Co-produced by INED and INSEE, the Trajectoires et Origines 2 (TeO2) survey, collected in 2019-2020, directly builds up on the TeO1 survey (2008)(2009). It follows the main principles of the first edition: 60% of the questions are identical and the sampling strategy is similar [Beauchemin et al.] The TeO2 questionnaire provides information on the migration history of the respondents and/or their parents, describes their educational and professional backgrounds, their family history, their marital life, their children, their housing conditions, their health, and the transmission of languages and religion. It comprehensively examines the access of individuals to social resources (school, work, housing, services, health care, etc.) as well as the discriminations that may hinder it. The scope of TeO2 is individuals aged 18 to 59 living in ordinary housing in metropolitan France. The survey was carried out among approximately 27,200 people, with the aim of executing detailed analyses of the main population groups that have direct or indirect experience of migration to metropolitan France. Immigrants and overseas natives, as well as their children born in metropolitan France, were therefore over-represented. The sample also includes individuals representative of the rest of the population. Since the TeO1 sample did not provide adequate coverage of the descendants of immigrants aged 50-59, the temporal comparisons made here only concern those aged 18-49.


The Interaction of Polyphenol Flavonoids with β-lactoglobulin: Molecular Docking and Molecular Dynamics Simulation Studies

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Saber El
email: saber.el-arem@polytechnique.edu

Shearing effects on the breathing mechanism of a cracked beam section in bi-axial flexure

Keywords: beam finite element, breathing crack, unilateral contact, shearing effects, convex analysis

 by taking into account the effects of shearing in the constitutive equations of a beam cracked section in bi-axial flexure. The paper describes the derivation of a lumped cracked beam model from the three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. Properties of the potential energy and convex analysis are used to reduce the three-dimensional computations needed for the model identification, and to derive the final form of the elastic energy that determines the nonlinear constitutive equations of the cracked transverse section. We aim to establish a relation of behavior between the applied forces and the resulting displacements field vectors, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis. The approach has been applied to the case of a cracked beam with a single crack covering the half of its circular cross section.

1 Introduction: state of the art Since the early 1970s when investigations on the vibrational behavior of cracked rotors began, numerous papers on this subject have been published, as a literature survey by [START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF] shows. The analysis of the behavior of cracked rotating machinery shafts is a complex structural problem. It requires, for a relevant description, a fine and precise modeling of the shaft and cracks in order to allow the identification and calculation of the parameters characterizing their presence. Researchers dealing with the problem of rotating cracked beams recognize its two main features, namely:

• the determination of the local flexibility of the beam cracked section;

• the consideration of the opening-closing phenomenon of the crack during the shaft rotation, commonly called breathing mechanism of the crack, and responsible of the system nonlinear behavior: when the shaft is rotating, then the crack opens and closes according to the stresses developped in the cracked surface. If these stresses are extensive, then the crack opens, resulting in a reduced shaft stiffness. When the stresses are compressive, then the crack remains closed and the shaft has the same stiffness as the non-cracked shaft. Thus, the system stiffness is depending on the cracked section position (Figure 1).

This breathing mechanism depends on the shaft rotation in the case when the static deflection dominates the vibration of the rotating shaft. This is a very common situation in large turbine-generators rotors. During the last three decades, great attention has been paid by several research scientists to the analysis and diagnosis of cracks in rotating machinery. The excellent review papers by [START_REF] Entwistle | Survey of the use of vibration methods in the assessment of component geometry[END_REF], [START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF], Wauer [1990a] and [START_REF] Gasch | A survey of the dynamic behavior of a simple rotating shaft with a transverse crack[END_REF] cover many aspects of this area and summarize the most relevant analytical, experimental and numerical works conducted in the last three decades and related to the cracked structures modeling.

There have been different attempts to quantify local effect introduced by cracks. The analysis of the local flexibility of a cracked region of structural element was quantified in the 1950s by Irwin [1957a,b], [START_REF] Bueckner | The propagation of cracks and the energy of elastic deformation[END_REF], [START_REF] Westmann | Stress analysis of cracked rectangular beams[END_REF] by relating it to the Stress Intensity Factors (SIF). Afterwards, the efforts to calculate the SIF for different cracked structures with simple geometry and loading was duplicated [START_REF] Tada | The stress analysis of cracks handbook[END_REF][START_REF] Duong | Mécanique de la rupture fragile[END_REF].

Most researchers agree with the application of the linear fracture mechanics theory to evaluate the local flexibility introduced by the crack [START_REF] Gross | Stress intensity factors for a single-edge-notch specimen in bending or combined bending and tension by boundary collocation of a stress function[END_REF][START_REF] Anifantis | Identification of peripheral cracks in cylindrical shells[END_REF][START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF][START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF], Papadopoulos and Dimarogonas, 1987a,b,c, Papadopoulos, 2004]. Obviously, the first work was done in the early 1970s by [START_REF] Dimarogonas | Dynamic response of cracked rotors[END_REF][START_REF] Dimarogonas | Dynamics of cracked shafts[END_REF] and [START_REF] Pafelias | Dynamic response of a cracked rotor[END_REF] at the General Electric Company. The energy release rate approach combines the linear fracture mechanics to rotordynamics theory in order to calculate the compliance caused by a transverse surface crack affecting a rotating shaft. A good review on this method is presented by [START_REF] Papadopoulos | The strain energy release approach for modeling cracks in rotors: A state of the art review[END_REF].

For an elastic structure, the additional displacement u due to the presence of a straight crack of depth a under the generalized loading P is given by the Castigliano theorem

u = ∂ ∂P a 0 G(a)da (1)
G is the energy release rate related to the SIF by the Irwin formula [Irwin, 1957a]. Then, the local flexibility matrix coefficients are obtained by

c ij = ∂ 2 ∂P i ∂P j a 0 G(a) da, 1 ≤ i, j ≤ 6 (2)
Extra diagonal terms of this matrix are responsible for longitudinal and lateral vibrations coupling that could be with great interest when dealing with cracks detection.

In two technical notes of the NASA, Gross andSrawley [1964, 1965] computed the local flexibility corresponding to tension and bending including their coupling terms. This coupling effect was observed by [START_REF] Rice | The part-through surface crack in an elastic plate[END_REF] in their study of cracked elastic plates for stress analysis.

Dimarogonas and his co-workers [START_REF] Dimarogonas | Crack identification of in aircraft structures[END_REF][START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF][START_REF] Dimarogonas | A fuzzy logic neutral network structured expert system shell for diagnosis and prognosis-users manual[END_REF][START_REF] Dimarogonas | A general purpose rotor dynamic analysis program-users manual[END_REF], and [START_REF] Anifantis | Identification of peripheral cracks in cylindrical shells[END_REF] introduced the full (6 × 6) flexibility matrix of a cracked section. They noted the presence of extra diagonal terms which indicate the coupling between the longitudinal and lateral vibrations. Papadopoulos and Dimarogonas [1987a,b,c], and [START_REF] Ostachowicz | Coupled torsional and bending vibrations of a rotor with an open crack[END_REF] computed all the (6 × 6) flexibility matrix of a Timoshenko beam cracked section for any loading case. However, there are no results for the SIF for cracks on a cylindrical shaft. Thus, [START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF] have developed a procedure which is commonly used in FEM software: the shaft was considered as an assembly of elementary rectangular strips where approximation of the SIF using fracture mechanics results remains possible (Figure 2). The SIF are obtained by integration of the energy release rate on the crack tip by

c ij = ∂ 2 ∂P i ∂P j a 0 b -b G(a, ξ) dξ da (3)
Although it offers the advantage of being easy to insert in a numerical algorithm, this method has some limitations. In fact, some numerical problems were reported by [START_REF] Abraham | Remark on the determination of compliance coefficients at the crack section of a uniform beam with circular cross section[END_REF] when the depth of the crack a exceeds the section radius R. [START_REF] Dimarogonas | Author's reply[END_REF], in his reply, stated that this divergence does not reflect reality; it is due to the assumption of a 2D stress distribution which is not valid near the ends of the crack tip. [START_REF] Papadopoulos | Coupling of bending and torsional vibration of a cracked timoshenko beam[END_REF] suggested to consider equation (3) with :

0.90b ≤ |ξ| ≤ 0.95b
when the crack depth a exceeds the shaft radius R. Wauer [1990b] explored the dynamics of a cracked, distributed parameter rotor component. The proposed model is a rotating Timoshenko shaft which is also flexible in extension and torsion. The stiffness matrix is constructed using the energy release rate approach as described in papers by Papadopoulos and Dimarogonas [1987a,b], and [START_REF] Gudmundson | The dynamical behavior of slender structures with crosssectional cracks[END_REF]. The geometric discontinuity due to the crack is replaced by a load discontinuity. The procedure reduces the problem to equations for one uniform beam with a modified load distribution. The proposed approach is a powerful instrument to obtain approximate results by a relatively small calculation expense. Although the shearing is considered in this approach, the author did not discuss the importance of its effects on the crack breathing mechanism.

2 Review of the Andrieux and Varé model

An original method for deriving a lumped model for a cracked beam section was proposed by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]. Based on three-dimensional computations, the procedure incorporates more realistic behavior of the cracks than the previous models, namely the unilateral contact conditions on the crack lips and their breathing mechanism under variable loading. The method was derived from three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. The authors established properties of the potential energy of this problem to reduce the amount of computation required for its determination in the case of a beam containing cracks of any shape and number. Convex analysis was also used to derive the final form of the energy that determines the nonlinear constitutive equations of the section of the beam which was incorporated in a FE analysis code. Great attention is paid to the capability of such a model to take into account the real 3D geometry of a crack and to represent, as general as possible, the effects of different load components on its nonlinear behavior.

The experimental validation of the approach is presented by [START_REF] Stoisser | A comprehensive theoretical, numerical and experimental approach for crack detection in power plant rotating machinery[END_REF]. The authors reported that the model reproduces with good accuracy the overall behavior of the shaft line in presence of cracks. The experimental validation allows the use of the model, with confidence and reliability, for the determination of the dynamics of supposed cracked rotors.

Based on the approach of [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF]Maitournam [2008, 2007] presented a method of construction of a cracked beam finite element which they used afterwards for the stability analysis of cracked shafts.

The authors distributed the additional energy due to the cracked section on the entire length of the cracked beam finite element. Considerable gain in computing efforts was reached compared to the nodal representation of the cracked section when dealing with the numerical integration of differential equations in structural dynamics.

The cracked structure of Figure 3(a) is subjected to an end moment M 2L = (M x (2L), M y (2L)) at z = 2L. [START_REF] Andrieux | Détermination de la loi de comportement en flexion bi-axée d'une poutre fissurée avec prise en compte du contact sur les fissures[END_REF] has demonstrated certain properties of the problem elastic energy, W * , leading to a considerable gain in the three-dimensional calculus required for the identification of the constitutive equations. In particular, for a linear elastic material, under the small displacements and small deformations assumptions, and in the absence of friction on the crack lips, the energy function could be written by distinguishing the contribution of the cracked section from that of the non-cracked elements, in the form :

W * (M 2L ) = W * (M ) = W * s (M ) + w * (M ) = L EI ||M || 2 (1 + s(Φ)) (4)
where M = (M x , M y ) is the resulting couple of flexural moments at the cracked section, W * s the total elastic energy of non-cracked structure subjected [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] to the flexural moment M and w * the additional elastic energy due to the presence of the cracked section. In order to simulate a rotating load on a fixed beam, the bending load is applied in several aperture angles Φ = atan( My Mx ), with Φ varying over [0, 2π[. E is the Young modulus and I quadratic moment of inertia. In this framework, the nonlinear constitutive equations of the discreet element modeling the cracked section are obtained by differentiating w * with respect to M . We obtain

[θ] =    [θ x ] [θ y ]    = 2L EI    s(Φ) -1 2 s ′ (Φ) 1 2 s ′ (Φ) s(Φ)       M x M y    with s ′ (Φ) = ds(Φ) dΦ (5)
However, for finite element computational codes in rotordynamics, a nonlinear relation of the form [θ] = f (M ) is to be identified. Thus, [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] introduced some properties of the additional deformation energy due to the cracked section, w([θ]). Thus, w could be written as a quadratic function of the rotations jumps as:

w([θ x ], [θ y ]) = EI 4L k(ϕ)||[θ]|| 2 with ϕ = atan( [θ y ] [θ x ] ) (6)
The Légendre-Fenchel transform is used to establish the relation between the two energy functions w * and w, then the stiffness function k is obtained from the compliance function s identified from three-dimensional calculus.

As described by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], the constitutive equations are finally obtained by differentiating w with respect to [θ] as

   M x M y    = EI 2L    k(ϕ) -1 2 k ′ (ϕ) 1 2 k ′ (ϕ) k(ϕ)       [θ x ] [θ y ]    with k ′ (ϕ) = dk(ϕ) dϕ (7)
The purpose of this paper is to complete the models suggested by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] and El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement (in french)[END_REF] by considering the shearing effects in the constitutive equations of a cracked beam section in bi-axial flexure. Our main purpose is to establish constitutive relations (nonlinear) between the applied forces and the resulting displacements field, which is compatible with the beams theory in order to allow the model exploitation for shafts dynamics analysis using a beam model.

Description of the current model

The study of the in-plane flexural behavior of a cracked beam by El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement (in french)[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF] showed that for straight line tip cracks the shearing and flexure effects could be separated when aiming to identify the constitutive equations of a cracked transverse section. In fact, the bending moments are only responsible for the rotations discontinuities, and the shear forces for the displacements discontinuities (slips). In this paper, we present the case of the bi-axial flexural behavior.

Our objective is to identify the characteristics of a one-dimensional model (Figure 3 In the one-dimensional (beam) model, the non-cracked parts of the structures are represented by Timoshenko beam elements, and the cracked section by a nodal element (zero length) allowing discontinuities of displacements and rotations. Indeed, this nodal element is composed of two uncoupled nonlinear springs modeling the cracked section flexural and shearing stiffnesses, cf. Figure 4. The unilateral contact between the crack lips is considered.

The following assumptions are considered for the modeling of the structure: • assumption of small displacements and small deformations,

• linear and isotropic elastic material,

• transverse cracks of any shape and in any number,

• unilateral contact without friction between the crack lips,

• the crack is completely closed in the unstressed configuration.

Properties of the energies functions of the problem

Let W * be the total elastic energy of the system. According to what precedes, W * can be put in the form

W * (F ) = W * s (F ) + w * (F ) = W * s (F ) + w * f (F ) + w * c (F ) (8)
where W * s denotes the total elastic energy of the non-cracked structure under the loading F (2L), and w * (F ) the elastic energy due to the presence of the crack. The study of the in-plane flexural behavior showed that the bending and shearing effects can be dissociated. Based on this result, w * is divided into: According to [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement (in french)[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF], w * is strictly convex and positively homogeneous of degree 2, from which rise the following properties:

• w * f (F ) = w * f (M )
Property 1 Functions w * c and w * f are strictly convex.

Property 2 Functions w * c and w * f are positively homogeneous of degree 2 :

∀λ ≥ 0 we have :

     w * f (λM ) = λ 2 w * f (M ) w * c (λT ) = λ 2 w * c (T ) (9)
It should be noticed that an essential hypothesis for obtaining properties 1 and 2 is that the gap between the lips of the crack is zero in the unstressed configuration. For the contribution of the bending moments, the quadratic form of w * f used in the study presented by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], is considered:

w * f (M ) = L EI ||M || 2 s(Φ) (10)
With Φ = atan( My(L) Mx(L) ) and s(Φ) the additional flexibility due to the presence of the crack and related to the effects of flexural moments at the cracked section of the beam. Energy due to the shearing effects at the cracked section is also quadratic and can be put in the form:

w * c (T ) = L µκS ||T || 2 s c (Φ c ), with Φ c = atan( T y (L) T x (L) ) (11)
µ is the shear modulus and κ the shear correction factor of Timoshenko. Due to property 2, the problem of identification of the function w * c on R 2 is reduced to the identification of the flexibility function s c (Φ c ) on the interval [0, 2π] by considering the particular case of ||T || = 1. This additional flexibility is identified from three-dimensional finite element calculus using Code Aster c . [START_REF] Varé | Calcul d'un modèle de rotor fissuré pour le code cadyro[END_REF] presented a simpler form of w * c . They brought the problem back, for certain shapes of cracks, to the identification of three constants by writing:

w * c (T ) = 1 2 T 2 x (L)s x + 1 2 T 2 y (L)s y + s xy T x (L)T y (L) (12)
s x , s y and s xy depend only on the geometry of the crack, they are independent of the parameters of loading and identified using the three following threedimensional finite element calculations cases:

w * c (T x = 1, T y = 0), w * c (T x = 0, T y = 1) and w * c (T x = 1, T y = 1)
This model allowed the obtaining, for certain cracks geometries, of results in excellent agreement with the three-dimensional FE calculations. However, it cannot be applied to any geometry of crack affecting beams or shafts with various cross section forms. Indeed, the shearing forces at the cracked section lead to the opening and closure of the crack according to mode II and/or mode III (modes of fracture). This depends on the orientation of the shearing force defined here by angle Φ c . The contact area between the lips of the crack, and consequently the additional flexibility due to its presence, depends on the direction of the applied shear force.

The present method, based on energy properties, could, without the use of additional simplifying assumptions, be applied to cracks of any shape and number affecting the same beam transverse section.

Constitutive equations of the discrete element with shearing effects consideration

The required relation describing the nodal element (Figure 4(d)) behavior is of the form:

     T M      = f (      [u] [θ]      ) (13)
with

[u] =      [u x ] [u y ]      =      u x (L + ) -u x (L -) u y (L + ) -u y (L -)      and [θ] =      [θ x ] [θ y ]      =      θ x (L + ) -θ x (L -) θ y (L + ) -θ y (L -)     
Consequently, it is the stiffness matrix of the nodal element which would be necessary to identify. Thus, we need to exploit the properties of the additional elastic deformation energy w due to the presence of the cracked section for the displacements discontinuities [u] and the rotations discontinuities

[θ].
Given the assumption of separating the shearing and flexure effects at the cracked section, w could be written in the form:

w([u], [θ]) = w f ([θ]) + w c ([u]) (14)
w f and w c are the elastic deformation energies associated with flexure and shearing, respectively. According to the work by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF], w f and w c have the following properties:

Property 3 Functions w c and w f are strictly convex.

Property 4 Functions w c and w f are positively homogeneous of degree 2 :

∀λ ≥ 0 we have :

     w f (λ[θ]) = λ 2 w f (λ[θ]) w c (λ[u]) = λ 2 w c ([u]) (15)
For w f , the form proposed by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] is considered:

w([θ x ], [θ y ]) = EI 4L k(ϕ)||[θ]|| 2 (16) With ϕ = atan( [θ y ] [θ x ]
) and k the flexural stiffness function of the nodal element.

Function k is deduced from s by using the Légendre-Fenchel transform relating w f to w * f . Following the development steps described in [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], [START_REF] Varé | Calcul d'un modèle de rotor fissuré pour le code cadyro[END_REF]) for w f , w c can be put in the form :

w c ([u]) = µκS 4L ||[u]|| 2 k c (ϕ c ) (17)
With ϕ c = atan( [uy] [ux] ) and k c the shearing stiffness function of the nodal element to be identified. Using the Légendre-Fenchel transform we establish the relation between w c to w * c by writing:

w c ([u]) = sup T (T • [u] -w * c (T )) = sup ||T ||=1 λ≥0 (λT • [u] -w * c (λT )) = sup ||T ||=1 λ≥0 (λT • [u] -λ 2 L µκS ||T || 2 s c (Φ c )) (18)
Contrary to the flexural flexibility function s of the nodal element which vanishes for certain couples of flexural moments leading to a total closing of the crack, the linearity noted with respect to the shearing action shows that the flexibility function s c due to shearing is never zero:

∀T = 0, we have [u] = 0
Thus, the required maximum in ( 18) is reached when:

λ =      µκS 2L T •[u] sc(Φc) if T • [u] ≥ 0 0 otherwise (19)
This leads to:

k c (ϕ c ) = sup ||T ||=1 T •[u]≥0 ( (T • [u]) 2 s c (Φ c ) ) (20) Thus, for [u] =      cosϕ c sinϕ c      and T =      cosΦ c sinΦ c     
, we obtain:

k c (ϕ c ) = sup cos(Φc-ϕc)≥0 ( cos 2 (Φ c -ϕ c ) s c (Φ c ) ) (21)
The condition cos(Φ cϕ c ) ≥ 0 implies, for a given ϕ c , that:

Φ c ∈ [ϕ c - π 2 , ϕ c + π 2 ]
The relation ( 21) becomes :

k c (ϕ c ) = sup Φc∈[ϕc-π 2 ,ϕc+ π 2 ] ( cos 2 (Φ c -ϕ c ) s c (Φ c ) ) (22)
The constitutive equations of the cracked section is obtained by differentiating w([u], [θ]) as follows:

             T x = ∂w([u],[θ]) ∂[ux] = ∂wc([u]) ∂[ux] , T y = ∂w([u],[θ]) ∂[uy] = ∂wc([u]) ∂[uy] M x = ∂w([u],[θ]) ∂[θx] = ∂w f ([θ]) ∂[θx] , M y = ∂w([u],[θ]) ∂[θy] = ∂w f ([θ]) ∂[θy] (23)
Finally, the required relation is given by:

          T x T y M x M y           =                  µκS 2L k c (ϕ c ) -µκS 4L k ′ c (ϕ c ) 0 0 µκS 4L k ′ c (ϕ c ) µκS 2L k c (ϕ c ) 0 0 0 0 EI 2L k(ϕ) -EI 4L k ′ (ϕ) 0 0 EI 4L k ′ (ϕ) EI 2L k(ϕ)                            [u x ] [u y ] [θ x ] [θ y ]          
(24)

Identification of functions s c and k c

Three-dimensional finite element calculations were carried out under the Code aster c to identify s c . The cracked structure, cf. Figure 4(a), is then subjected, at z = 2L, to the loading vector:

F = T x (2L) = cos(Φ c ), T y (2L) = sin(Φ c ), M x (2L) = LT y (2L), M y (2L) = -LT x (2L)
which leads, at the cracked section (z = L), to:

(T , M ) = T x (L), T y (L), M x (L), M y (L) = T x (2L), T y (2L), 0, 0 Thus, the additional deformation energy due to the cracked section w * is reduced to w * c . The angle Φ c varies in [0 • , 360 • [ at a rate of a loading case every Since ||T || = 1, and using the relation ( 11), the additional shearing flexibility function is obtained, for all loading cases, by:

s c (Φ c ) = µκS w * c (T ) L
Figure 5 shows the evolution of s c for a straight line front crack with a relative depth a D = 50%. It is noticed in particular that s c does not vanish on [0, 2π[. Also, the strong dependency of s c on the loading direction Φ c is clearly visible. Moreover, we show that the model presented by [START_REF] Varé | Calcul d'un modèle de rotor fissuré pour le code cadyro[END_REF] corresponds to particular loading cases since

s x = 2L µκS s c (Φ c = 0), s y = 2L µκS s c (Φ c = π 2 ) and s xy = 2L µκS s c (Φ c = π 4 )
The stiffness function k c is then calculated by using the formula (22), cf. Figure 6(a). Finally, its derivative k ′ c is calculated using the centered differences method, cf. Figure 6(b).

Validation of the approach

Now that we have identified the constitutive equations of the nodal element that represents the cracked transverse section, it is the objective of this section to introduce it in a one-dimensional FE code and see if we get sufficient agreement with three-dimensional representation. For a better validation of the approach, it is certainly more adequat to consider a cracked structure different from the one used in the identification procedure of the nodal element.

Let consider a cylinder of axis (oz), diameter D = 0.5m, quadratic moment of inertia I = πD 4 64 , total length 3m, clamped in z = 0 and subjected at the other end to a couple of forces

T = (T x = cos(α), T y = sin(α)) with α ∈ [0, 2π[
The cylinder contains, at z = 1m, a straight line front crack with a relative depth a D = 50%. Three-dimensional calculations accounting for the unilateral contact without friction between the lips of the crack are carried using the code aster c . The one-dimensional model of the same structure is made of two non-cracked elements of Timoshenko beam type and respective lengths 1m and 2m connected at z = 1m by a nodal element that represents the cracked section ( node 2), cf. Figure 7. Two cases are considered:

• in the first, the effects of shearing at the cracked section are neglected. Only discontinuities of rotations are then allowed at node 2 ([u x ] = [u y ] = 0). In this case we consider the model presented by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]. • in the second, the nodal element behavior is described by relation (24): shearing effects are considered in this case.

Figure 8 shows an excellent agreement between the beam and the threedimensional models. It appears that the effects of shearing on the crack breath- ing mechanism are negligible since the [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] model (with no consideration of the shearing effects) and the current model (taking into account the shearing effects) give the same results. The breathing mechanism is, thus, governed by the normal stresses on the crack lips.

Conclusions

The present approach consists in introducing the effects of shearing in the constitutive equations of a beam cracked transverse section. Three-dimensional calculations were carried out in order to identify the required nonlinear relations. In particular, the 3D nonlinear finite element calculations allowed the breathing mechanism of the crack to be predicted accurately. Based on the properties of the problem energy, the procedure presented in this paper could be applied to cracks of any shape and, moreover, to the case of multiple cracks affecting the same section. The approach was applied to the case of a single straight line front crack covering the half of a cylindrical beam cross section ( a D = 50%). For this form of crack, as illustrated in the Figure 8, the shearing effects on the breathing mechanism of the crack are negligible when compared to those of the flexural moments. The opening and closure of the crack are governed by the normal stress on the crack lips. The relative slip between the crack lips associated with shear forces remains very small compared to the rotations jumps due to the flexural moments at the cracked section (Figure 9).

Fig. 1 .

 1 Fig. 1. Crack breathing mechanism

Fig. 2 .

 2 Fig. 2. Geometry of the shaft cracked section

  Fig.3. The cracked beam model of[START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]

  (c) ) from accurate three-dimensional FE model (Figure3(a)). The transverse cracked section (Figure3(b)) is replaced by two lumped nonlinear flexural and shearing springs (Figure3(d)) for which we aim to establish the constitutive equations. The three-dimensional FE model considered is that of a cylinder of axis (oz), diameter D = 1m, cross section S, length 2L = 4D, containing, at midspan, a cracked section, cf. Figure4. The structural element, clamped at its end z = 0, is subjected at z = 2L to a couple of shear forces and a couple of flexural moments: (T x (2L), T y (2L), M x (2L), M y (2L)).

Fig. 4 .

 4 Fig. 4. The current cracked beam modeling

  denotes the part of w * (F ) due to the resulting couple of flexural moments M = (M x (L), M y (L)) at the cracked section, and • w * c (F ) = w * c (T ) the part due to the couple of shear forces T = (T x (L), T y (L)).

Fig. 5 .Fig. 6 .

 56 Fig. 5. Additional shearing flexibility s c , a D = 50%

Fig. 8 .

 8 Fig. 7. Structure Modelling: 3D and beam Finite elements

Fig. 9 .

 9 Fig. 9. discontinuties of the displacements and rotations at the cracked section, a D = 50%.

Acknowledgements The author would like to thank Quoc-Son Nguyen and Habibou Maitournam (LMS, Ecole Polytechnique, France) for very helpful discussions.


Saber El

Arem Habibou Maitournam
email: maitournam@lms.polytechnique.fr

A cracked beam finite element for rotating shaft dynamics and stability analysis

Keywords: breathing crack, beam, unilateral contact, finite element, rotor, Floquet, Stability

. Based on this flexibility which is distributed over the entire length of the element, a cracked beam finite element stiffness matrix is deduced. Considerable gain in computing efforts is reached compared to the nodal representation of the cracked section when dealing with the numerical integration of differential equations in structural dynamics. The stability analysis of a cracked shaft is carried using the Floquet theory.

Introduction

Because of the increasing need of energy, the plants installed by electricity supply utilities throughout the world are becoming larger and more highly stressed. Thus, the risk of turbogenerator shaft cracking is increasing also. Rotating shafts are omnipresent in Aeronautics, aerospace, automobile industries and in particular in the energy sector which is vital for any economic development. Fatigue cracks is an important form of rotor damage which can lead to catastrophic failures if undetected early. They can have detrimental effects on the reliability of rotating shafts. According to [START_REF] Bently | Detection of rotor cracks[END_REF], in the 70s and till the beginning of the 80s, at least 28 shaft failures due to cracks were registered in the USA energy industry. Thus, Since the 80s, the interest of researchers to characterize structures containing cracks is growing remarkably. Between the beginning of the 70s and the end of the 90s, more than 500 articles concerning the cracked structures was published [Dimarogonas, 1996, Bachschmid and[START_REF] Bachschmid | Crack effects in rotordynamics[END_REF].

The problem of determination of the behavior of cracked structures has been tackled with for a long time. And the fact that a crack presence or a local defect in a structural member introduces a local flexibility that affects its vibration response was known long ago. This local flexibility is related to the strain energy concentration in the vicinity of the crack.

The study of cracked turbines rotors began in the industry of the energy in the USA with the works of [START_REF] Dimarogonas | Dynamic response of cracked rotors[END_REF][START_REF] Dimarogonas | Dynamics of cracked shafts[END_REF]. In Europe, the first works appeared only some years later in papers by [START_REF] Gasch | Dynamical behavior of a simple rotor with a cross-sectional crack[END_REF], [START_REF] Mayes | The vibrational behaviour of a rotating shaft system containing a transverse crack[END_REF] and [START_REF] Henry | Vibrations in cracked shafts[END_REF]. These authors considered a simple model to account for the crack breathing mechanism to which we often make reference by "the switching crack model " or "the hinge crack model" : the crack is totally opened or totally closed.

The vibration analysis of cracked beams or shafts is a problem of great interest due to its practical importance. In fact, vibration measurements offer a non-destructive, inexpensive and fast means to detect and locate cracks. Thus, vibration behavior analysis and monitoring of cracked rotors has received considerable attention in the last three decades [START_REF] Gudmundson | Eigenfrequency changes of structures due to cracks, notches or other geometrical changes[END_REF][START_REF] Gudmundson | The dynamical behavior of slender structures with crosssectional cracks[END_REF]. It has, perhaps, the greatest potential since it can be carried without dismantling any part of the machine and usually online avoiding the costly downtime of turbomachinery. [START_REF] Zuo | Etude du comportement dynamique des systmes linéaires par morceaux-contribution la détection des fissures dans les arbres de machines tournantes[END_REF], and [START_REF] Zuo | Nonlinear real and complex modes of conewise linear systems[END_REF] also used a bilinear model to characterize the vibrational response of a cracked shaft with the aim of developing an online cracks detection method. They, first, examined the 1 degree of freedom (dof) system then studied the behavior of a system with 2 dof. By extending the Rosenberg normal mode notion for smooth and symmetric nonlinear systems to conwise linear systems, they defined the nonlinear modes of the bilinear system which were calculated numerically, and in certain simple cases, analytically. The application of this method to systems with high number of dof is complicated and would lead to high computation costs.

Bachschmid and his co-workers [START_REF] Bachschmid | The influence of unbalance on cracked rotors[END_REF][START_REF] Bachschmid | Identificationof multiple faults in rotor systems[END_REF][Bachschmid et al., , 2004b] ] examined the effects of the presence of a crack on the vibratory response of a rotor or a pump axis. Experimental and numerical models were proposed, the thermal effects on the crack breathing mechanism were taken into account [Bachschmid et al., 2004b]. It was reported that the temperature distribution is not influenced by the presence of the crack unlike those of the stresses and deformations.

A comparison of various cracked beams models was presented by [START_REF] Friswell | Crack modelling for structural health monitoring[END_REF]. The authors showed that in the low frequencies domain, simple models of the crack breathing mechanism and beam type elements are adequate to monitor structures health. However, the approach of modeling based on a bilinear dynamic system, which still makes reference, remains a simplistic approach which leads to some reserves about the quality of the quantitative results stemming from its exploitation.

A good review on the most relevant analytical, experimental and numerical works conducted in the three last decades and related to the cracked structures behavior were reported by [START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF], [START_REF] Entwistle | Survey of the use of vibration methods in the assessment of component geometry[END_REF], [START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF], [START_REF] Wauer | On the dynamics of cracked rotors: A literature survey[END_REF], [START_REF] Gasch | A survey of the dynamic behavior of a simple rotating shaft with a transverse crack[END_REF], El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF].

Today, most of the works on the cracked shafts vibration analysis are concerned with investigating more deeply certain particular points such as the phase of acceleration or deceleration of the shaft, the passage through the critical speed or the coupling between diverse modes of vibrations to highlight parameters facilitating the online cracks detection when dealing with machines monitoring. These works such as those of [START_REF] Darpe | Transient response and breathing behaviour of a cracked jeffcott rotor[END_REF], those of [START_REF] Jun | Modelling and vibration analysis of a simple rotor with a breathing crack[END_REF] and those of Sinou andLees [2005, 2007] where we find numerical and experimental results of their investigations on certain points such as aforementioned, remain faithful to the theoretical principles formalized in the 70s in some reference papers. In the other hand, remarkable and continuous progress of the computational tools allows the realization of successful three-dimensional modelings. So, it becomes possible to envisage an identification of the law of behavior of a cracked shaft section (breathing mechanism description) which frees from certain simplifying hypotheses and approximations made until now; this without degrading the costs of the calculations of the vibrational response.

The main objective of this work is the presentation of a method of construction of a cracked beam finite element. The stiffness variation of the element is deduced from three-dimensional finite element computations accounting for the unilateral contact between the cracks lips. Based on an energy approach, this method could be applied to cracks of any shape. The validation of the approach on a case of a cracked rotating shaft is then presented and its stability analysis is carried using the Floquet theory. The work of El [START_REF] Arem | Un élément fini de poutre fissurée: application la dynamique des arbres tournants (in french)[END_REF] is taken back to show the general form of the stiffness matrix of the finite element and then facilitate the approximation of its terms.

Cracked rotors modeling: State of the art

The analysis of rotating machinery shafts behavior is a complex structural problem. It requires, for a relevant description, a fine and precise modeling of the rotor and cracks in order to allow the identification and calculation of the parameters characterizing their presence.

Researchers dealing with the problem of rotating cracked beam recognize its two main features, namely:

• the determination of the local flexibility of the beam cracked section;

• the consideration of the crack breathing mechanism responsible of the system nonlinearity: the system stiffness is depending on the cracked section position.

Most researchers agree with the application of the linear fracture mechanics theory to evaluate the local flexibility introduced by the crack [START_REF] Gross | Stress intensity factors for a single-edge-notch specimen in bending or combined bending and tension by boundary collocation of a stress function[END_REF][START_REF] Anifantis | Identification of peripheral cracks in cylindrical shells[END_REF][START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF][START_REF] Dimarogonas | Vibration of cracked structures: A state of the art review[END_REF], Papadopoulos and Dimarogonas, 1987a,b,c, Papadopoulos, 2004]. Obviously, the first work was done in the early 1970 by [START_REF] Dimarogonas | Dynamic response of cracked rotors[END_REF][START_REF] Dimarogonas | Dynamics of cracked shafts[END_REF] and [START_REF] Pafelias | Dynamic response of a cracked rotor[END_REF] at the General Electric Company. There have been different attempts to quantify local effect introduced by cracks. The analysis of the local flexibility of a cracked region of structural element was quantified in the 1950s by Irwin [1957a,b], [START_REF] Bueckner | The propagation of cracks and the energy of elastic deformation[END_REF] , [START_REF] Westmann | Stress analysis of cracked rectangular beams[END_REF] by relating it to the Stress Intensity Factors (SIF). Afterwards, the efforts to calculate the SIF for different cracked structures with simple geometry and loading was duplicated [START_REF] Tada | The stress analysis of cracks handbook[END_REF][START_REF] Duong | Mécanique de la rupture fragile[END_REF].

For an elastic structure, the additional displacement u due to the presence of a straight crack of depth a under the generalized loading P is given by the Castigliano theorem

u = ∂ ∂P a 0 G(a)da (1)
G is the energy release rate defined in fracture mechanics and related to the SIF by the Irwin formula [Irwin, 1957a]. Then, the local flexibility matrix coefficients are obtained by

c ij = ∂ 2 ∂P i ∂P j a 0 G(a) da, 1 ≤ i, j ≤ 6 (2)
Extra diagonal terms of this matrix are responsible for longitudinal and lateral vibrations coupling that can be with great interest when dealing with cracks detection.

In two technical notes of the NASA, Gross andSrawley [1964, 1965] computed the local flexibility corresponding to tension and bending including their coupling terms. This coupling effect was observed by [START_REF] Rice | The part-through surface crack in an elastic plate[END_REF] in their study of cracked elastic plates for stress analysis.

Dimarogonas and his co-workers [START_REF] Dimarogonas | Crack identification of in aircraft structures[END_REF][START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF][START_REF] Dimarogonas | A fuzzy logic neutral network structured expert system shell for diagnosis and prognosis-users manual[END_REF][START_REF] Dimarogonas | A general purpose rotor dynamic analysis program-users manual[END_REF], and [START_REF] Anifantis | Identification of peripheral cracks in cylindrical shells[END_REF] introduced the full (6 × 6) flexibility matrix of a cracked section. They noted the presence of extra diagonal terms which indicate the coupling between the longitudinal and lateral vibrations. Papadopoulos and Dimarogonas [1987a,b,c], and Ostachowicz and Krawwczuk [1992] computed all the (6 × 6) flexibility matrix of a Timoshenko beam cracked section for any loading case.

However, there are no results for the SIF for cracks on a cylindrical shaft. Thus, [START_REF] Dimarogonas | Analytical methods in rotor dynamics[END_REF] have developed a procedure which is commonly used in FEM software: the shaft was considered as an assembly of elementary rectangular strips where approximation of the SIF using fracture mechanics results remains possible. The SIF are obtained by integration of the energy release rate on the crack tip. Although it offers the advantage of being easily applicable in a numerical algorithm, this method remains an approximation whose convergence remains to be checked. In fact, some numerical problems were noted [START_REF] Abraham | Remark on the determination of compliance coefficients at the crack section of a uniform beam with circular cross section[END_REF][START_REF] Dimarogonas | Author's reply[END_REF] when the depth of the crack exceeds the section radius. Moreover, generalization of this method to any geometry of cracked section is complex, even impossible in the case of non connate multiple cracks affecting the same transverse section. An original method for deriving a lumped cracked section beam model was proposed by [START_REF] Varé | Modélisation d'une section de poutre fissuréeapplication aux rotors de turbine[END_REF] and [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF]. The procedure was designed by starting from three-dimensional computations and incorporating more realistic behavior on the cracks than the previous models, namely the unilateral contact conditions on the cracks lips and the breathing mechanism of the cracks under variable loading. The approach was validated experimentally by [START_REF] Audebert | Comportement dynamique de rotors avec fissuration transverse: modélisation et validation expérimentale[END_REF] and applied for the study of real cracked structures specially turbines.

The cracked element of Figure 1 is submitted to an end moment M 2L = (M x (2L), M y (2L)) at z = 2L. [START_REF] Andrieux | Détermination de la loi de comportement en flexion bi-axée d'une poutre fissurée avec prise en compte du contact sur les fissures[END_REF] has demonstrated certain properties of the problem elastic energy , W * , leading to a considerable gain in three-dimensional calculus needed for the identification of the behavior law.

In particular, for a linear elastic material, under the small displacements and small deformations assumptions, and in the absence of friction on the cracks lips, the energy function could be written by distinguishing the contribution of the cracked section from that of the uncracked elements, in the form :

W * (M 2L ) = W * (M ) = W * s (M ) + w * (M ) = L EI ||M || 2 (1 + s(Φ)) (3)
where M = (M x , M y ) is the resulting couple of flexural moments at the cracked section , W * s (M ) the total elastic energy of uncracked element submitted to the flexural moment M and w * (M ) the additional elastic energy due to the presence of the cracked section. The loading direction angle is defined by Φ = atan( M y Mx ). E is the Young modulus and I quadratic moment of inertia. In this framework, the nonlinear behavior law of the discreet element modelling the cracked section is obtained by derivation of the function w * (M ) by M . We obtain

[θ] =    [θ x ] [θ y ]    = 2L EI    s(Φ) -1 2 s (Φ) 1 2 s (Φ) s(Φ)       M x M y    with s (Φ) = ds(Φ) dΦ (4)
However, for finite element computational codes in rotordynamics, the compliance function s(Φ) is of low interest and a nonlinear relation of the form [θ] = f (M ) is to be integrated in transient computations. Thus, [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF] introduced some properties of the additional strain energy due to the cracked section, w ([θ]). Thus, w could be written as a quadratic function of the rotations jumps as:

w([θ x ], [θ y ]) = EI 4L k(ϕ)||[θ]|| 2 with ϕ = atan( [θ y ] [θ x ] ) (5)
By using the Légendre-Fenchel transform to establish the relation between the two energy functions w * (M ) and w([θ]), the stiffness function k(ϕ) is obtained from the compliance function s(Φ) identified from three-dimensional calculus.

As described by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF], the behavior law is finally deduced from the derivation of w ([θ]) by [θ] as

   M x M y    = EI 2L    k(ϕ) -1 2 k (ϕ) 1 2 k (ϕ) k(ϕ)       [θ x ] [θ y ]    with k (ϕ) = dk(ϕ) dϕ (6) z Tx1 Mx1 A Ty1 My1 My2 Tx2 Mx2 Ty2 B Le Le
Fig. 2. The cracked beam finite element.

A cracked beam finite element construction

There are two procedures to introduce the local flexibility generated by a cracked section when dealing with the numerical integration of differential equations in dynamics. The first technique considers the construction of a stiffness matrix exclusively for the cracked section by computing the inverse of the flexibility matrix. However, for small cracks, the additional flexibility is very small and, consequently, the corresponding stiffness coefficient are extremely large leading to high numerical integration costs and convergence problems [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF]].

The second procedure, adopted here, consists in the construction of a cracked finite element stiffness matrix, which is later assembled with the other uncracked elements of the system. Thus, the elastic energy due to the cracks is distributed over the entire length of the cracked element. This method has been used in works by Bachschmid et al. [2004a] and [START_REF] Saavedra | Crack detection and vibrational behavior of cracked beams[END_REF].

The studies of [START_REF] Verrier | Calcul d'un modle de rotor fissuré avec prise en compte de l'effort tranchant-cas de la flexion plane[END_REF], El [START_REF] Arem | Loi de comportement en flexion d'une section de poutre fissurée avec prise en compte des effets de cisaillement (in french)[END_REF], [START_REF] Varé | Modélisation d'une section de poutre fissurée en flexion: prise en compte des efforts tranchants (in french)[END_REF] and El [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF] showed that the shear effects on the breathing mechanism of the cracks is insignificant and will be neglected in this study.

Let consider the cracked finite element of length 2L e , circular transverse section of diameter D and quadratic moment of inertia I, cf. Figure 2.

First, we clamp all the displacements of node A and establish a relation of the form

u = S(f ) • f (7) f = {T x 2 , T y 2 , M x 2 , M y 2 } t and u = {u x 2 , u y 2 , θ x 2 , θ y 2 } t denote
, respectively, the loading and displacements vectors at the end section (z = 2L e ). S(f ) represents the compliance matrix of the structure. At the cracked section (z = L e ), the internal efforts are given by :

                   T x = T x 2 T y = T y 2 M x = M x 2 -L e T y M y = M y 2 + L e T x (8)
The breathing mechanism of the cracks is governed by the flexural moment direction Φ = atan( M y Mx ) at the cracked section. The elastic energy of the cracked element could be written as

W * (f ) = W * s (f ) + w * (M ) = W * s (f ) + L EI ||M || 2 s(Φ) (9)
W * s (f ) denotes the elastic energy of the uncracked finite element of the same geometry and submitted to the same loading conditions. By using equation (3), the additional elastic energy due to the cracked section is given by

w * (M ) = L EI ||M || 2 s(Φ) ( 10
)
where L is the half length of the three-dimensional element used to identify the compliance function s(Φ) as described below.

The nonlinear relation between the applied efforts and the resulting displacements at the end section (z = 2L e ) are obtained by derivation of the function W * by f . Thus, by using (8), we write

u = S(f ) • f = S(Φ) • f (11)
where

S(Φ) = S 0 + 2L EI           L 2 e s(Φ) -L 2 e 2 s (Φ) L e 2 s (Φ) L e s(Φ) L 2 e 2 s (Φ) L 2 e s(Φ) -L e s(Φ) Le 2 s (Φ) -L e 2 s (Φ) -L e s(Φ) s(Φ) -1 2 s (Φ) L e s(Φ) -L e 2 s (Φ) 1 2 s (Φ) s(Φ)           (12)
S 0 denotes the compliance matrix of an uncracked beam element of length 2L e . Let consider {u B/A } the relative displacement of node B in rapport of node A. It verifies the relation

{T x 2 , T y 2 , M x 2 , M y 2 } t = (S(Φ)) -1 {u B/A } (13)
From equilibrium conditions of the element of Figure 2, the internal forces in B can be expressed in terms of those in A as:

                   T x 1 = -T x 2 T y 1 = -T y 2 M x 1 = -M x 2 + 2L e T y 2 M y 1 = -M y 2 -2L e T x 2 (14)
or, written in a matrix form:

{T x 1 , T y 1 , M x 1 , M y 1 , T x 2 , T y 2 , M x 2 , M y 2 } t = Π 1 {T x 2 , T y 2 , M x 2 , M y 2 } t (15) with Π 1 =                          -1 0 0 0 0 -1 0 0 0 2L e -1 0 -2L e 0 0 -1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1                          (16)
In addition, when writing {u B/A } in the form

{u B/A } = {u 1 B/A , u 2 B/A , u 3 B/A , u 4 B/A } we obtain                    u x 2 = u 1 B/A + u x 1 + 2L e θ y 1 u y 2 = u 2 B/A + u y 1 -2L e θ x 1 θ x 2 = u 3 B/A + θ x 1 θ y 2 = u 4 B/A + θ y 1 (17)
or, in a matrix form

{u B/A } = Π 2 {u x 1 , u y 1 , θ x 1 , θ y 1 , u x 2 , u y 2 , θ x 2 , θ y 2 } t ( 18
)
where

Π 2 =           -1 0 0 -2L e 1 0 0 0 0 -1 2L e 0 0 1 0 0 0 0 -1 0 0 0 1 0 0 0 0 -1 0 0 0 1           (19)
Comparing with equation ( 16), it can be seen that:

Π 2 = Π t 1
Moreover, the cracked beam finite element stiffness matrix K ef verifies

{T x 1 , T y 1 , M x 1 , M y 1 , T x 2 , T y 2 , M x 2 , M y 2 } t = K ef {u x 1 , u y 1 , θ x 1 , θ y 1 , u x 2 , u y 2 , θ x 2 , θ y 2 } t (20)
Substituting equation ( 15) into equation ( 20) gives

Π 1 {T x 2 , T y 2 , M x 2 , M y 2 } t = K ef {u x 1 , u y 1 , θ x 1 , θ y 1 , u x 2 , u y 2 , θ x 2 , θ y 2 } t (21)
Then, substituting equation ( 13) into equation ( 21), results in

Π 1 (S(Φ)) -1 {u B/A } = K ef {u x 1 , u y 1 , θ x 1 , θ y 1 , u x 2 , u y 2 , θ x 2 , θ y 2 } t (22)
Finally, using equation ( 18) leads to:

K ef = Π 1 (S(Φ)) -1 Π t 1 (23)
In this relation, the stiffness matrix appears as depending on the loading efforts represented by angle Φ. However, in a finite element code, it is preferable to express relation ( 23) as a function of the problem's unknowns, that is the nodal displacements. We start by writting :

(S(Φ)) -1 = K(ϕ e ) = K 0 -K e (ϕ e ) (24)
to distinguish the stiffness matrix of an uncracked element of length 2L e and inertial moment I, Π 1 K 0 Π t 1 , from the matrix modelling the cracked section presence Π 1 K e (ϕ e )Π t 1 . ϕ e is the angle given by ϕ e = atan(

θy 2 -θy 1 θ x 2 -θ x 1
) and K 0 by [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF]

K 0 = S 0 -1 = EI 2L e (1 + a)            3 L 2 e 0 0 -3 Le 0 3 L 2 e 3 Le 0 0 3 Le 4 + a 0 -3 L e 0 0 4 + a            (25) a = 12EI
4µkSL 2 e is the shearing effects coefficient. For an Euler-Bernoulli beam element, a is zero.

Equation ( 24) leads to

K e (ϕ e ) = K 0 -(S(Φ)) -1 = EI 2L           0 0 0 0 0 0 0 0 0 0 k xx (ϕ e ) k xy (ϕ e ) 0 0 k yx (ϕ e ) k yy (ϕ e )           ( 26
)
where

             k xx (ϕ e ) = k yy (ϕ e ) = L 2 (4L e s(Φ)+4Ls 2 (Φ)+Ls 2 (Φ)) L e (4L 2 e +8LL e s(Φ)+4L 2 s 2 (Φ)+L 2 s 2 (Φ)) k xy (ϕ e ) = -k yx (ϕ e ) = - 2L 2 s (Φ) 4L 2 e +8LLes(Φ)+4L 2 s 2 (Φ)+L 2 s 2 (Φ) (27)
When L = L e , we obtain

             k xx (ϕ e ) = k yy (ϕ e ) = 4s(Φ)+4s 2 (Φ)+s 2 (Φ) 4+8s(Φ)+4s 2 (Φ)+s 2 (Φ) k xy (ϕ e ) = -k yx (ϕ e ) = - 2s (Φ) 4+8s(Φ)+4s 2 (Φ)+s 2 (Φ) (28)
Using the three-dimensional calculus conducted on the cracked element of Figure 1, we identify the compliance function s(Φ) as described previously. Then, we determine the stiffness matrix K e (ϕ e ) terms by using relation (26).

We have noticed that: Thus, K e (ϕ e ) can be written in the form:

k xy (ϕ e ) = - 1 2 k xx (ϕ e ) (29)
K e (ϕ e ) = EI 2L           0 0 0 0 0 0 0 0 0 0 k xx (ϕ e ) -1 2 k xx (ϕ e ) 0 0 1 2 k xx (ϕ e ) k xx (ϕ e )           (30)
For a straight crack of depth a = D 2 , these terms are shown by Figure 3. We notice small variations on [0, In order to validate the stiffness beam finite element matrix construction method presented above, we propose to compare the three-dimensional modeling calculus results to those obtained using a beam modeling of the cracked structure of Figure 4. The cylinder element of axis (oz), of diameter D = 0.5m, and total length L t = 3m, is clamped at its end z = 0 and submitted at the other to couple of efforts T = (T x = cos(α), T y = sin(α)) with α varying from 0 to 2π. The cracked section is located at z = 1 m. The crack is straight with depth a = 0.5D, cf. Figure 4(a). The three-dimensional finite element calculus take into account the unilateral contact conditions between the crack lips. The beam model consists of two beam finite elements: the first one is a cracked beam finite element of length 2 m. The second is a classic beam finite element of length 1 m, cf. Figure 4(b). Figure 5 shows excellent results concordance .

Vibratory response of a cracked shaft

This section is deserved to explore the vibratory response of a shaft with a cracked section under own weight effects. The system is composed of a beam of distributed mass m, circular section S and diameter D = 0.20 m. The cracked section is located at mid-span. The structure is simply supported at node 1 and node 6 and submitted to the effects of its own weight. The shaft is rotating at the frequency Ω about (oz) axis. The system is divided into five elements of length l = 4D, cf. Figure 6. The elements 1 -2, 2 -3, 4 -5 and 5 -6 represent the structure uncracked parts. 3 -4 is a cracked beam finite element.

At t = 0, the crack is totally open, cf. Figure 6(b). We suppose that the stiffness matrix remains constant between two instants t n = nh and t n+1 = (n + 1)h where h is the time step used for the numerical integration of the dynamical system. For the validity of this approximation, the time steps should be relatively small compared to the excitation period ( 1 Ω ). In the low frequency domain, this difficulty is easily overcame. Thus, the stiffness matrix is updated at the end of each integration step. The HHT method is adopted for the numerical integration of the obtained system [START_REF] Hilber | Improved numerical dissipation for time integration algorithms in structural dynamics[END_REF] with:

α = 1 3 , γ = 1 2 + α et β = 1 4 (1 + α) 2
which corresponds, in the linear analysis, to an unconditionally stable scheme with maximum precision. With this modeling, the unilateral contact conditions between the crack lips are taken into account exactly. In fact, when the crack is totally closed, we obtain K e (ϕ e ) = 0 and the cracked element stiffness matrix is the one of an uncracked element. Overmore, the time step used here is 10 -3 s which is at least 20 to 100 times smaller than the ones used for penalization-implicit approach when the cracked section is modeled by a nodal element (element of length zero). In this later approch, the time step depends on the value given to the penalization constant and is always less than 10 -5 s to ensure calculus convergence.

The vibratory response shows the superharmonic resonance phenomena presence when the rotating frequency passes through entire divisions of the critical speed w 1 . Thus, for ξ ≈ w 1 n with n ∈ N the shaft orbit and the phase diagram are composed of n interlaced loops (cf.

Figure 7 where the viscous damping d is of 5%). Also, the vibratory amplitude of harmonic n× reaches, at this rotating frequency, higher levels, cf. Figure 8.

Stability analysis

The stability of a cracked shaft is analyzed using the Floquet method Floquet [1879], [START_REF] Nayfeh | Coupled torsional and bending vibrations of a rotor with an open crack[END_REF], [START_REF] Nayfeh | Applied nonlinear dynamics[END_REF]. This method was used by [START_REF] Gasch | Dynamical behavior of a simple rotor with a cross-sectional crack[END_REF], [START_REF] Meng | Stability ans stability degree of a cracked flexible rotor supported on journal bearings[END_REF] and El [START_REF] Arem | A simple model for the dynamical behavior of a cracked rotor[END_REF] for the stability analysis of a two parameters cracked rotating shaft. The first step of this section consists of approximating the K e (ϕ e ) terms by a classical function of the crack depth a and angle ϕ e . We have noticed that the function k xx (ϕ e ) for different straight tip crack depths shows that it could be approximated by:

k xx (ϕ e ) ≈ k max sin 4 ( ϕ e 2 - π 4 )e (sin( ϕ e 2 -π 4 )) 4 (31)
where k max is given by

k max ≈ 3.43( a D ) 2.73 (32)
These approximations remain satisfactory for cracks of depth going up to 30 percent of the diameter of the shaft, cf. Figure 9.

By using the Floquet theory, we have investigated, numerically, the stability of a cracked rotating shaft with a straight tip crack at mid-span, cf. Figure 6. Results, cf. Figure 10, show three principal instability areas: the first zone corresponds to (0 < ξ < 0.5) superharmonic resonance phenomenon. The second is located around the exact resonance (ξ = 1) and the third area (around ξ = 2) corresponds to subharmonic resonance. It's important to note that even for weak viscous damping (d ≈ 1 % ) the stability of the cracked shaft is only slightly affected. The zones of instabilities appear for Ω near the first critical speed (ξ ≈ 1) and twice the critical speed (ξ ≈ 2) and correspond to deep cracks ( a D > 25% ).

Conclusions

A method for the construction of a cracked beam finite element is presented.

The crack breathing phenomenon is finely described since the flexibility due to the presence of the cracks is deduced from three-dimensional finite element calculations taking into account of the conditions of unilateral contact between the cracks lips as originally developed by [START_REF] Andrieux | A 3d cracked beam model with unilateral contactapplication to rotors[END_REF].

The precise descriptions of the loss of stiffness and of the progressive closure or opening of the cracks are of fundamental importance.

The approach is quite simple and comprehensive and can be applied to any geometry of cracks. It is important to note the considerable gain in computational efforts when comparing with the use of the technique of penalization.

Indeed this technique, used when the cracked section is modeled by a nodal element [START_REF] Arem | Vibrations non-lineaires des structures fissurées: application aux rotors de turbines[END_REF], leads to the appearance of very high numerical frequencies (without physical signification). The time steps considered for the temporal numerical integration of the dynamic system are then very small and the computation costs are, consequently, very important. Compared to the nodal representation of the cracked section and the technique of penalization to consider the conditions of contact between the cracks lips, this model has the following advantages:

• taking into account in an exact way of the phenomenon of cracks breathing mechanism, • reduction of the costs of calculations from 20 to 100 times.

In the study of the cracked shafts, the researchers often supposed, to reduce the difficulty of the problem, that the amplitude of the vibrations due to the presence of cracks is weak compared to those of the vibrations due to permanent loads (El [START_REF] Arem | A simple model for the dynamical behavior of a cracked rotor[END_REF], [START_REF] Gasch | Dynamical behavior of a simple rotor with a cross-sectional crack[END_REF], [START_REF] Mayes | The vibrational behaviour of a rotating shaft system containing a transverse crack[END_REF], [START_REF] Mayes | A method of calculating the vibrational behaviour of coupled rotating shafts containning a transverse crack[END_REF], [START_REF] Davies | The vibrational behaviour of multi-shaft, multi-bearing system in the presence of the propagating transverse crack[END_REF], [START_REF] Henry | Vibrations in cracked shafts[END_REF], [START_REF] Gasch | A survey of the dynamic behavior of a simple rotating shaft with a transverse crack[END_REF]). The present model is more general and allows to overcome limitations due to such an assumption.

The stability analysis of the cracked shaft of Figure 6 was carried using the Floquet theory. Figure 10 shows stable and instable zones in the (ξ, a D ) plan for different viscous damping coefficients. We have noticed that the instability zones disappear for d ≥ 3%. It can be deduced that, for real turbines shafts, where the viscous damping is about 3% to 4% [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF], the effects of a cracked section at mid-span on the stability of the structure is negligible when a D is less than 35% .

Fig. 1 .

 1 Fig. 1. The cracked beam element modeling

Fig. 3 .

 3 Fig. 3. K e (ϕ e ) terms for a straight crack of depth a = D 2 , L e = L = 2D.

Fig. 4 .

 4 Fig. 4. Three-dimensional and beam modeling of the system.

Fig. 5 .

 5 Fig. 5. Model comparison to three-dimensional results and nodal modelling, a D = 0.50.

  Fig. 6. Finite element modeling of the cracked structure.

Fig. 7 .

 7 Fig. 7. Examples of node 2 orbits, a D = 0.50, d=0.05.

Fig. 8 .

 8 Fig. 8. Examples of node 3 u(t) amplitude spectra, a D = 0.50, d=0.05.

Fig. 9 .Fig. 10 .

 910 Fig. 9. k xx (ϕ e ) approximation for different straight crack depths.

Acknowledgements The authors would like to thank Huy Duong Bui for useful discussions.


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1 SubmissiontoMedicalSociologyOnline,4August2009,733words.

HealthPolicyandPolitics

AlisonHann(ed) Ashgate,2007. ISBN9780754670643.206pp.£55.00(hbk). http://www.ashgate.com/default.aspx?page=637&edition_id=10402 The topic of health policy-making in Britain seems inextinguishable, constantly renewedbyshiftsinthebalanceofpowerbetweenitsmainplayers-thestate,the

  medicalprofessionandcivilsociety.Inthatcontext,thepaperscollectedinHealth PolicyandPoliticsexaminethepoliticalfactorsthatcurrentlyshapethemakingof health policyintheNationalHealthService(NHS).Itsfourteenchapterscovera widearrayofoverarchingissuesthataffectvirtuallyallformsoftreatmentwithin the health care sector, except for two more specific case studies (Chapter 10 on 'protracted waiting for orthopaedic services' and Chapter 12 on immunization policy).Mostchaptersaredrawnfromconferencepapersgivenbymembersofthe HealthPoliticsGroupbetween2005and2006. An important transversal theme of the book relates to the design and reform of structural,system-levelaspectsoftheNHS,whichAlisonHannintroducesasthe 'macro'levelofBritishhealthpolicy-making.Chapter2exploresthemechanisms thathelpexplainingacharacteristicparadoxoftheNHS,orasRudolfKleinonce put it, a paradox of famine among plenty: 'how did a creditable record on NHS expenditureafter2002…cometoco-existwithrecorddeficits?'(p.14)NICE) face a different challenge as they strive to establish themselves as legitimate decision-making bodies in the NHS, a task that blameavoiding governments tend to avoid to handle themselves because, as shown in Chapter7,of'thehighpoliticalprofileenjoyedbytheNHS,leadingtotheprospect oflostvotesforanypartyseentodamageit'(p.93). Thebookinevitablyleadsthereadertolookforwardtothemostimportantfactors that might guide future developments of health policy. Alongside mentions of European regulatory convergence over particular aspects of health care, such as druglicensing(Chapter13,p.174),twochaptersareparticularlyinterestinginthat respect. Chapter 1 provides a brief survey of the impact of European regulations, derived mostly from common market agreements, on public health policy-a trend,onemayhint,whichissettoincreaseinthecomingyears.Aparallelsource ofinfluenceoverhealthpolicy-makingstemsfromtheriseofa'NewPublicHealth' ideology described in minute detail and through a careful examination of its historicallineageinChapter9the third sector in supporting state-led treatment facilities and biomedical research,orthegovernanceofthenursingprofessionintheNHS,alltopicswhich becomeevenmoresalientwhenobservedincomparativesettings.

Reviewer:

FrançoisBriatte InstituteofPoliticalStudies UniversityofGrenoble f.briatte@ed.ac.uk