Language Treasure Trove: Duoxu Vocabularies and Texts by Wu Wancai

From 2013 to 2017 three ELDP grantees; Katia Chirkova, Han Zhengkang, and Wang Dehe worked on the documentation and description of a critically endangered Tibeto-Burman language, Duoxu, as part of the "Comparative and Cross-Varietal Documentation of Highly Endangered Languages of South-West China" (MDP0257) project, funded by the Endangered Languages Documentation Programme. The research team's documentation work consisted of recording the speech of the last fluent Duoxu speakers, supplemented by vocabulary lists collected by Duoxu community members. One manuscript in particular, compiled by Wu Wancai (1926Wancai ( -2006)), allowed the research team to acquire a more complete picture of the language and to even tentatively reconstruct some vocabulary items that could no longer be recalled by the few remaining speakers. Hence, Wu Wancai's manuscript enriched our understanding of Duoxu and helped us to create a more representative documentation record of this disappearing language.

-----Duoxu (/do³³-ɕu³³-na³¹/, Duōxù 多续 or Duōxū 多须 in Mandarin Chinese) is a Tibeto-Burman language, spoken in Mianning county (冕宁县), which is located in the Liangshan Yi Autonomous Prefecture (凉山彝族自治州) in Sichuan province (四川省) in the People's Republic of China. Duoxu is part of a close-knit cluster of three languages (Duoxu, Lizu, Ersu), which due to their close genetic relationship, and despite their mutual unintelligibility, are currently classified as three dialects of one Ersu language (ISO-639 code ers).

Duoxu stands out from the many minority languages of China for two reasons; first, it is among the most critically endangered languages in the country. Stigmatized and virtually forbidden during the Cultural Revolution (1966)(1967)(1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975)(1976), Duoxu went out of use by the end of the 1970s only to be spoken in household situations. Currently, hardly any of the 2,000 Duoxu people speak the traditional language of their community. A survey of all remaining Duoxu speakers in April 2013, conducted as part of the aforementioned ELDP project identified a mere nine remaining speakers. Sadly, due to their elderliness and age-related health problems, only two speakers (Mr. Wu Rongfu 伍荣富 and Mrs. Wu Decai 吴德才) were able to participate in language documentation work. Given that background, our documentation work faced considerable difficulties. On the one hand, the speech of the last speakers we were working with was characterized by extensive individual variation. In addition, as none of the speakers had the chance of using Duoxu in everyday life (where they normally use the Mianning dialect of Southwest Mandarin instead), they typically could not recall even some very basic words, such as common animal names, bird names, and traditional household utensils. On the other hand, in the absence of previous attestations of Duoxu dating from before the language went out of use, we lacked external sources to evaluate the intactness and consistency of the data we collected with the last speakers.

The spectacular discovery of several Duoxu vocabulary lists and texts, collected by individual community members and circulating in the Duoxu community, provided us with a means to solve some of the above problems. One manuscript in particular stood out for the sheer amount of data it contained: a lengthy vocabulary list of over 2700 words and a collection of texts, including song lyrics, proverbs, riddles, and narratives. , a native Duoxu speaker. Worried about the fate of his language, Wu started collecting Duoxu data in 1955 and worked on his manuscript for over 47 years. To our knowledge, his manuscript provides the largest collection of lexical data on Duoxu known to date, which furthermore stem from the time when the language was still actively spoken.

2. Front page of the manuscript 3. Photo of Wu Wancai Using Wu's data was not without challenges. One complication was that the data were transcribed in Chinese characters, which at best approximate the pronunciation of words in languages other than Chinese. Another complication is that, in addition to regular Chinese characters, Wu created a number of phonograms in order to record Duoxu consonant-vowel combinations that do not occur in Chinese (such as nja or ki; some examples of regular characters and Wu's self-created phonograms are provided on the following page from the manuscript, self-created phonograms are encircled). Unfortunately, Wu did not leave any indications of how those phonograms were to be read. To be able to use the rich Duoxu data in Wu's manuscript, our research team conducted an additional study of the Mianning dialect of Southwest Mandarin, which was Wu's native language. Having recorded all characters used by Wu in his Duoxu transcriptions, we could then systematically compare Wu's character transcriptions in Mianning Mandarin with modern Duoxu readings of words from Wu's list that could be recalled by and independently recorded with our Duoxu language consultants. We were thereby able to establish regular correspondence patterns between transcriptions and modern Duoxu readings and to get an understanding of the basic principles underlying Wu's transcription. That, in turn, provided a key to interpreting the phonograms created by Wu.

Examples of regular and self-created characters

By combining both a regular documentation work with the last speakers and an analysis of community records of the language, we could a richer and more diverse picture of Duoxu, and brought back to life long-forgotten words, expressions, and even entire narratives.

An annotated version of Wu Wancai's word list and some selected stories, accompanied by phonetic transcriptions of their Mianning Duoxu readings and modern Duoxu readings is in preparation. In the meantime, photos of the manuscript pages, manuscript-based elicitation sessions with our Duoxu consultants, and many more audio and video recordings of Duoxu can be consulted here.

  1. Photo of the main Duoxu consultants Wu Rongfu (伍荣富, left) and Wu Decai (吴德才, right), November 2017, photo by Katia Chirkova Second, Duoxu is among the least studied Tibeto-Burman languages of China. There are no published data on Duoxu from before the language went out of use, and prior to our project, the entire documentation record of Duoxu consisted of two short lexical lists and one basic grammatical sketch.

  That manuscript was entitled Liangshan zhou Zangzu jianshi: Mianning xian Zangzu Duoxu zhi pu 凉山州藏族简史:冕 宁县藏族多续之谱 [A brief history of the Tibetans of Liangshan Prefecture: The Duoxu Tibetans of Mianning county]. It was compiled by Wu Wancai 吴万才


Publication of the First-Ever Textbook of Duoxu -----Katia Chirkova and Han Zhengkang, two ELDP grantees, have written the first-ever textbook of Duoxu, accompanied by sound files on a CD. The textbook was funded by ELDP (MDP 0257) and the Sichuan Research Institute for the Nationalities (四川省民族研究所).

-----Duoxu (/do³³-ɕu³³-na³¹/, Duōxù 多续 or Duōxū 多须 in Mandarin Chinese) is a littleknown and virtually undescribed Tibeto-Burman language, spoken in Mianning county ( 冕宁县), which is located in the Liangshan Yi Autonomous Prefecture (凉山彝族自治州) in Sichuan province (四川省) in the People's Republic of China. Together with Lizu and Ersu, two closely related languages, Duoxu is currently classified as a member of the Qiangic subgroup of the Tibeto-Burman language family. Duoxu is critically endangered. It is spoken by no more than a few members of the oldest generation, who do not use the language on a regular basis and who, for that reason, have a relatively limited knowledge of it. In order to document as much as possible of this language, it formed part of the "Comparative and Cross-Varietal Documentation of Highly Endangered Languages of South-West China" (MDP 0257) project, which was funded by the Endangered Languages Documentation Programme (ELDP) and lasted from 2013 to 2017.

Early on in the project, in April 2013, the four investigators Katia Chirkova (p.i.), Yuan Xiaowen, Han Zhengkang, and Wang Dehe conducted a survey of all remaining Duoxu speakers. The survey identified nine last speakers, all in their 70s and 80s. All nine speakers are bilingual in the local variety of Southwest Mandarin, which is their dominant daily language, and none of them has regular conversation partners in Duoxu. These circumstances qualify the remaining nine Duoxu speakers as 'semi-speakers.' During our contact with the Duoxu speakers, they expressed interest in a textbook, so as to pass their knowledge of the language on to future generations, with a faint hope of reviving their moribund language. That is how we came up with the idea of writing a practical grammar, based on data collected by the research team of the documentation project. The resulting textbook is the first-ever comprehensive description of Duoxu. It presents the language data in the form of an introductory Duoxu course; outlining stepby-step the phonological system, which is a pinyin-based Romanization system for Duoxu, as well as the basic vocabulary, morphology, and syntactic patterns. All examples in the text are accompanied by sound files, provided on an accompanying CD. The aim of the book is to present the collected data in a form that would allow self-study of the language by members of the Duoxu Tibetan community, and others who are interested in learning this language. Katia Chirkova started working on the textbook in 2015, assisted by Han Zhengkang in the writing process. Yuan Xiaowen contributed the textbook's preface, and it was then published by Minzu Chubanshe 民族出版社 (Minorities Press) in Beijing in late 2016. In November 2017, after attending a conference in China, Katia Chirkova traveled to Mianning County, where she presented complementary copies of the textbook to the participants of the 2013 Duoxu language survey. The ceremony was organized by Wang Dehe, who also took pictures at the memorable occasion.

(

  To be included in the caption for the photo of the grammar): Katia Chirkova 齐卡佳 and Han Zhengkang 韩正康. 2016. Shiyong Duoxuyu Yufa 实用多 续语语法 [A Practical Grammar of Duoxu]. Beijing 北京: Minzu Chubanshe 民族出版社.


Sociolinguistic survey of Ersu

Sociolinguistic survey of Ersu

-----Katia Chirkova and Wang Dehe, two ELDP grantees, have conducted -in collaboration with James N. Stanford of Dartmouth College -a sociolinguistic survey of the Ersu language, aimed at investigating consonantal variation in this language. The survey was part of the "Comparative and Cross-Varietal Documentation of Highly Endangered Languages of South-West China" (MDP 0257) project, which was funded by the Endangered Languages Documentation Programme (ELDP) and lasted from 2013 to 2017. An article based on this sociolinguistic survey will be published in the forthcoming issue of the journal Language Variation and Change (30.1).

-----Ersu (/ə˞-s̪ v̩ ́ xò/, 尔苏语 ěrsūyǔ, ISO-639 code ers) is a little-studied Tibeto-Burman language spoken in rural areas of Southwest China. Together with Lizu and Duoxu, two closely related languages, Ersu is currently classified as a member of the Qiangic subgroup of the Tibeto-Burman language family. Ersu is endangered. Only half of the overall ethnic population of ca. 16,800 Ersu people are proficient speakers of Ersu. Moreover, Ersu is under increasing pressure from its neighbouring languages: Nuosu (or Yi, Lolo-Burmese, Tibeto-Burman) and Southwestern Mandarin Chinese (Sinitic).

Throughout history, the Ersu people were a traditional society, marrying strictly within their own group. The majority of people were monolingual in Ersu; and only a few Ersu people could speak languages of their ethnic neighbors. However, the past decades brought important changes to their lifestyle. Ersu-speaking areas became more closely integrated into the People's Republic as a nation, and as a result, the national language of Mandarin Chinese has become prestigious and dominant in the Ersu community. The rapid transition toward an unequal prestige relationship between Ersu and Mandarin has led to considerable variation and change in Ersu. The sociolinguistic survey, which is the topic of this blog post, aimed to study these processes.

Data for the survey were collected in two field trips to Ersu-speaking areas in February and August of 2015. They amounted to over 24 hours of speech data from 97 speakers of Ersu (52 women, 45 men, ages 8-94) with a focus on the following four variables:

(1) loss of typologically uncommon trilled retroflex sounds (e.g., [ The recorded data included word elicitation (a total of 116 words) and ethnographic interviews. All sound files, textgrids, transcriptions, and metadata related to this study are now made available at the Endangered Languages Archive (ELAR; http://elar.soas.ac.uk). They can be used for further exploration and acoustic analysis.

Quantitative analysis of the data reveals that language change is indeed in progress. The change we observed is phonological convergence, or the leveling of phonological differences between Ersu and its dominant contact language, Mandarin Chinese. The Ersu case study furthermore shows that convergence between languages in an unequal prestige relationship can be manifested in a socially stratified way, which is consistent with many of the classic Labovian sociolinguistic principles for age, sex, and social class.

To our knowledge, the present study is the first large-scale sociolinguistic survey of Ersu, and the first variationist study of any Tibeto-Burman language in Southwest China. It contributes to a more comprehensive documentation and description of the Ersu language and also to a better understanding of the process of phonological convergence and, more broadly, of variation and change in a multilingual setting.

An article by the research team outlining the results of the survey entitled "A long way from New York City: Socially stratified contact-induced phonological convergence in Ganluo Ersu (Sichuan, China)" is to appear in the forthcoming issue of the journal Language Variation and Change (30.1).

Captions for photos

  ʈɽ], [ɽ]) (2) delateralization of the lateral fricative ([ɬ] ~ [x]) (3) devoicing of stops and affricates (e.g., [b] ~ [p]) (4) simplification of onsets (e.g., [mb] ~ [b])


S I Dimitrov

Square-free numbers of the shapes x 2 + y 2 + z 2 + z + 1 and x 2 + y 2 + z + 1

Keywords: Square-free numbers, Asymptotic formula, Gauss sum, Salié sum. 2020 Math. Subject Classification: 11L05, 11N25, 11N37

In this paper we study the distribution of square-free numbers of the forms x 2 + y 2 + z 2 + z + 1 and x 2 + y 2 + z + 1. We establish asymptotic formulas for the number of triples of positive integers x, y, z ≤ H such that x 2 + y 2 + z 2 + z + 1 is square-free and respectively such that x 2 + y 2 + z + 1 is square-free.

Notations

Let H be a sufficiently large positive number. By ε we denote an arbitrary small positive number, not the same in all appearances. As usual µ(n) is Möbius' function, ω(n) denotes the number of distinct prime factors of n and τ (n) represents the number of positive divisors of n. Further [t] and {t} denote the integer part, respectively, the fractional part of t. Instead of m ≡ n (mod d) we write for simplicity m ≡ n (d). Moreover (l, m) is the greatest common divisor of l and m, and (l, m, n) is the greatest common divisor of l, m and n. The letter p will always denote prime number. We put t = min({t} , 1 -{t}). As usual e(t)=exp(2πit). For any odd q we denote by • q the Jacobi symbol. For any n and q such that (n, q) = 1 we denote by n q the inverse of n modulo q. Throughout this paper we shall denote the Gauss sums G(q, m, n) = q x=1 e mx 2 + nx q , G(q, m) = G(q, m, 0) ,

the Salié sum S(q, m, n) = q x=1 (x,q)=1

x q e mx + nx q q (2)

and the Kloosterman sum

K(q, m, n) = q x=1 (x,q)=1 e mx + nx q q . (3)
Define Γ(H) = 1≤x,y,z≤H

µ 2 (x 2 + y 2 + z 2 + z + 1) , (4)
Γ * (H) = 1≤x,y,z≤H

µ 2 (x 2 + y 2 + z + 1) , (5)
λ(q, l, m, n) = 1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) e lx + my + nz q ,

λ * (q, l, m, n) = 1≤x,y,z≤q x 2 +y 2 +z+1≡0 (q) e lx + my + nz q .

We define also λ(q) = λ(q, 0, 0, 0)

and λ * (q) = λ * (q, 0, 0, 0) .

2 Introduction and statement of the results

The problems for the existence of infinitely many square-free numbers of a special form are as interesting as well as difficult in multiplicative number theory. One of them is the representation of infinitely many square-free numbers by polynomials. There exist many articles that consider the k-free values of polynomials. In this connection in 2012 Tolev [START_REF] Tolev | On the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1 is squarefree[END_REF] showed that there exist infinitely many square-free numbers of the form x 2 + y 2 + 1. More precisely he established the asymptotic formula 1≤x,y≤H

µ 2 (x 2 + y 2 + 1) = p 1 - λ 2 (p 2 ) p 4 H 2 + O H 4 3 +ε , where λ 2 (q) = 1≤x,y≤q x 2 +y 2 +1≡0 (q) 1 .
Continuing these research Zhou and Ding [START_REF] Zhou | On the square-free values of the polynomial x 2 + y 2 + z 2 + k[END_REF] established that 1≤x,y,z≤H

µ 2 (x 2 + y 2 + z 2 + k) = p 1 - λ 3 (p 2 ) p 6 H 3 + O H 7 3 +ε , where λ 3 (q) = 1≤x,y,z≤q x 2 +y 2 +z 2 +k≡0 (q) 1 .
Recently, Chen and Wang [2] generalized the result of Zhou and Ding by deriving an asymptotic formula for the distribution of r-free numbers of the form x 2 + y 2 + z 2 + k. In the case r = 2, they obtained reminder term O H 3 +ε given by Zhou and Ding [START_REF] Zhou | On the square-free values of the polynomial x 2 + y 2 + z 2 + k[END_REF]. Another interesting problem is consecutive square-free numbers represented by polynomials. In 2020 the author [START_REF] Dimitrov | On the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1, x 2 + y 2 + 2 are square-free[END_REF] proved that there exist infinitely many square-free pairs of the type x 2 + y 2 + 1, x 2 + y 2 + 2 and showed that 1≤x,y≤H

µ 2 (x 2 + y 2 + 1) µ 2 (x 2 + y 2 + 2) = σ 2 H 2 + O H 8 5 +ε , (10)
where

σ 2 = p 1 - λ 2 (p 2 , 1) + λ 2 (1, p 2 ) p 4
and λ 2 (q 1 , q 2 ) = 1≤x,y≤q 1 q 2 x 2 +y 2 +1≡0 (q 1 ) x 2 +y 2 +2≡0 (q 2 ) 1 .

Subsequently the author [START_REF] Dimitrov | Pairs of square-free values of the type n 2 + 1, n 2 + 2[END_REF] proved that there exist infinitely many square-free pairs of the type x 2 + 1, x 2 + 2. Further, Jing and Liu [START_REF] Jing | Consecutive square-free numbers and square-free primitive roots[END_REF] improved the reminder term in [START_REF] Nathanson | Additive Number Theory : The Classical Bases[END_REF] to O H 3 2 +ε . Recently B. Chen [START_REF] Chen | On the consecutive square-free values of the polynomials x 2 1 +• • •+x 2 k +1[END_REF] generalized the results of the author and showed that

1≤x 1 ,...,x k ≤H µ 2 (x 2 1 + • • • + x 2 k + 1) µ 2 (x 2 1 + • • • + x 2 k + 2) = σ k H k + O H k-1 2 -1 2k +ε ,
where k ≥ 3,

σ k = p 1 - λ k (p 2 , 1) + λ k (1, p 2 ) p 2k and λ k (q 1 , q 2 ) = 1≤x 1 ,...,x k ≤q 1 q 2 x 2 1 +•••+x 2 k +1≡0 (q 1 ) x 2 1 +•••+x 2 k +2≡0 (q 2 )
1 .

Motivated by these results we continue to extend these studies by proving two theorems.

Theorem 1. For the sum Γ(H) defined by (4) the asymptotic formula

Γ(H) = p 1 - λ(p 2 ) p 6 H 3 + O H 9 4 +ε (11)
holds.

Theorem 2. For the sum Γ * (H) defined by (5) the asymptotic formula

Γ * (H) = p 1 - λ * (p 2 ) p 6 H 3 + O H 5 2 +ε (12)
holds.

3 Lemmas Lemma 1. For the Gauss sum we have

(i) If (q 1 , q 2 ) = 1 then G(q 1 q 2 , m, n) = G(q 1 , mq 2 , n) G(q 2 , mq 1 , n) . (ii) If (q 1 , q 2 ) = 1 then G(q 1 q 2 , m 1 q 2 + m 2 q 1 , n) = G(q 1 , m 1 q 2 2 , n) G(q 2 , m 2 q 2 1 , n) . (iii) If (q, m) = d then G(q, m, n) = d G (q/d, m/d, n/d) if d | n , 0 if d n . (iv) If (q, 2m) = 1 then G(q, m, n) = e -(4m) q n 2 q m q G(q, 1) . (v) If (q, 2) = 1 then G 2 (q, 1) = (-1) q-1 2 q .
Proof. See [START_REF] Estermann | A new application of the Hardy-Littlewood-Kloosterman method[END_REF] and [START_REF] Hua | Introduction to Number Theory[END_REF].

Lemma 2. Let q is odd integer. For the sum defined by (2) we have

|S(q, m, n)| ≤ 2 ω(q) √ q .
Proof. See [START_REF] Louvel | The first moment of Salié sums[END_REF].

Lemma 3. For the sum defined by (3) we have

|K(q, m, n)| ≤ τ (q) q 1 2 (q, m, n) 1 2 .
Proof. See [START_REF] Iwaniec | Analytic number theory[END_REF].

Lemma 4. Let (q 1 , q 2 ) = 1 . ( 13
)
For the function defined by (6) we have λ(q 1 q 2 , l, m, n) = λ q 1 , lq 2 q 1 , mq 2 q 1 , nq 2 q 1 λ q 2 , lq 1 q 2 , mq 1 q 2 , nq 1 q 2 .

Proof. On the one hand (1), ( 6), (13), the well-known formula

d k=1 e kn d = d , if d | n, 0 , if d n (14)
and Lemma 1 imply

λ(q 1 q 2 , l, m, n) = 1 q 1 q 2 1≤x,y,z≤q 1 q 2 e lx + my + nz q 1 q 2 1≤h≤q 1 q 2 e h(x 2 + y 2 + z 2 + z + 1) q 1 q 2 = 1 q 1 q 2 1≤h≤q 1 q 2 e h q 1 q 2 G(q 1 q 2 , h, l) G(q 1 q 2 , h, m) G(q 1 q 2 , h, n + h) = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 q 2 , h 1 q 2 + h 2 q 1 , l) G(q 1 q 2 , h 1 q 2 + h 2 q 1 , m) × G(q 1 q 2 , h 1 q 2 + h 2 q 1 , n + h 1 q 2 + h 2 q 1 ) = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 , h 1 q 2 2 , l) G(q 2 , h 2 q 2 1 , l) G(q 1 , h 1 q 2 2 , m) × G(q 2 , h 2 q 2 1 , m) G(q 1 , h 1 q 2 2 , n + h 1 q 2 + h 2 q 1 ) G(q 2 , h 2 q 2 1 , n + h 1 q 2 + h 2 q 1 ) = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 , h 1 q 2 2 , l) G(q 2 , h 2 q 2 1 , l) G(q 1 , h 1 q 2 2 , m) × G(q 2 , h 2 q 2 1 , m) G(q 1 , h 1 q 2 2 , n + h 1 q 2 ) G(q 2 , h 2 q 2 1 , n + h 2 q 1 ) . ( 15
)
On the other hand ( 1), ( 6), ( 13), ( 14) and Lemma 1 yield λ q 1 , lq 2 q 1 , mq 2 q 1 , nq 2 q 1 λ q 2 , lq 1 q 2 , mq 1 q 2 , nq

1 q 2 = 1 q 1 q 2 1≤x 1 ,y 1 ,z 1 ≤q 1 e lq 2 q 1 x 1 + mq 2 q 1 y 1 + nq 2 q 1 z 1 q 1 1≤h 1 ≤q 1 e h 1 (x 2 1 + y 2 1 + z 2 1 + z 1 + 1) q 1 × 1≤x 2 ,y 2 ,z 2 ≤q 2 e lq 1 q 2 x 2 + mq 1 q 2 y 2 + nq 1 q 2 z 2 q 2 1≤h 2 ≤q 2 e h 2 (x 2 2 + y 2 2 + z 2 2 + z 2 + 1) q 2 = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 , h 1 , lq 2 q 1 ) G(q 1 , h 1 , mq 2 q 1 ) G(q 1 , h 1 , h 1 + nq 2 q 1 ) × G(q 2 , h 2 , lq 1 q 2 ) G(q 2 , h 2 , mq 1 q 2 ) G(q 2 , h 2 , h 2 + nq 1 q 2 ) . ( 16
)
Using the substitution x → q 2 q 1 x we get

G(q 1 , h 1 q 2 2 , l) = q 1 x=1 e h 1 q 2 2 x 2 + lx q 1 = q 1 x=1 e h 1 x 2 + lq 2 q 1 x q 1 = G(q 1 , h 1 , lq 2 q 1 ) . (17)
Arguing in a similar way, we obtain

G(q 1 , h 1 q 2 2 , m) = G(q 1 , h 1 , mq 2 q 1 ) , (18)
G(q 1 , h 1 q 2 2 , n + h 1 q 2 ) = G(q 1 , h 1 , h 1 + nq 2 q 1 ) , (19)
G(q 2 , h 2 q 2 1 , l) = G(q 2 , h 2 , lq 1 q 2 ) , (20)
G(q 2 , h 2 q 2 1 , m) = G(q 2 , h 2 , mq 1 q 2 ) , (21)
G(q 2 , h 2 q 2 1 , n + h 2 q 1 ) = G(q 2 , h 2 , h 2 + nq 1 q 2 ) . (22)
Summarizing ( 15) -( 22) we complete the proof of the lemma.

Lemma 5. Let 8 q. For the function defined by (6) the upper bound λ(q, l, m, n) qτ (q)2 ω(q) (q, l, m, n)

holds. In particular we have λ(q, l, m, n) q 1+ε (q, l, m, n) and λ(q) q 2+ε .

Proof. Case 1. 2 q.

From (1), ( 2), ( 6), ( 14) and Lemma 1 we deduce λ(q, l, m, n)

= 1 q 1≤x,y,z≤q e lx + my + nz q 1≤h≤q e h(x 2 + y 2 + z 2 + z + 1) q = 1 q 1≤h≤q e h q G(q, h, l) G(q, h, m) G(q, h, h + n) = 1 q d|q 1≤h≤q (h,q)= q d e h q G(q, h, l) G(q, h, m) G(q, h, h + n) = q 2 d|q q d |(l,m,n) 1 d 3 1≤r≤d (r,d)=1 e r d G(d, r, ldq -1 ) G(d, r, mdq -1 ) G(d, r, r + ndq -1 ) = q 2 d|q q d |(l,m,n) G 3 (d, 1) d 3 1≤r≤d (r,d)=1 r d 3 e r -(4r) d r 2 + (l 2 + m 2 + n 2 )d 2 q -2 + 2rndq -1 d = q 2 d|q q d |(l,m,n) e - 2 d n q G 3 (d, 1) d 3 S d, 1 -4 d , -4 d (l 2 + m 2 + n 2 )d 2 q -2 . (23)
Now (23), Lemma 1 and Lemma 2 give us

λ(q, l, m, n) q 2 d|q q d |(l,m,n) d -3 d 3 2 d 1 2 2 ω(d) q 2 2 ω(q) q d |(q,l,m,n) d -1 q 2 2 ω(q)
r|(q,l,m,n)

q -1 r qτ (q)2 ω(q) (q, l, m, n) .

Case 2. q = 2 h q , where 2 q and h ≤ 2. Using (24), Lemma 4 and the trivial estimate |λ(2 h , l, m, n)| ≤ 8 h we find |λ(2 h q , l, m, n)| = λ 2 h , lq 2 h , mq 2 h , nq 2 h λ q , l2 h q , m2 h q , n2 h q 64q τ (q )2 ω(q ) q , l2 h q , m2 h q , n2 h q qτ (q)2 ω(q) (q, l, m, n) .

(

) 25
Now the lemma follows from (24) and (25).

Using Lemma 5 and arguing as in [START_REF] Zhou | On the square-free values of the polynomial x 2 + y 2 + z 2 + k[END_REF] we derive the following lemma.

Lemma 6. Assume that 8 q and H 0 ≥ 2. Then for the sums

Λ 1 = 1≤l≤H 0 |λ(q, l, 0, 0)| l , Λ 2 = 1≤n≤H 0 |λ(q, 0, 0, n)| n , Λ 3 = 1≤l,m≤H 0 |λ(q, l, m, 0)| lm , Λ 4 = 1≤l,n≤H 0 |λ(q, l, 0, n)| ln
and

Λ 5 = 1≤l,m,n≤H 0 |λ(q, l, m, n)| lmn the estimations Λ i q 1+ε H ε 0 , i = 1, 2, 3, 4, 5 hold. Lemma 7. Let (q 1 , q 2 ) = 1 . (26)
For the function defined by [START_REF] Iwaniec | Analytic number theory[END_REF] we have

λ * (q 1 q 2 , l, m, n) = λ * q 1 , lq 2 q 1 , mq 2 q 1 , nq 2 q 1 λ * q 2 , lq 1 q 2 , mq 1 q 2 , nq 1 q 2 .
Proof. On the one hand (1), ( 7), ( 14), (26) and Lemma 1 yield λ * (q 1 q 2 , l, m, n) = 1 q 1 q 2 1≤x,y,z≤q 1 q 2 e lx + my + nz q 1 q 2 1≤h≤q 1 q 2 e h(x 2 + y 2 + z + 1)

q 1 q 2 = 1 q 1 q 2 1≤h≤q 1 q 2 e h q 1 q 2 G(q 1 q 2 , h, l) G(q 1 q 2 , h, m) G(q 1 q 2 , 0, h + n) = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 q 2 , h 1 q 2 + h 2 q 1 , l) G(q 1 q 2 , h 1 q 2 + h 2 q 1 , m) × G(q 1 q 2 , 0, h 1 q 2 + h 2 q 1 + n) = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 , h 1 q 2 2 , l) G(q 2 , h 2 q 2 1 , l) G(q 1 , h 1 q 2 2 , m) × G(q 2 , h 2 q 2 1 , m)G(q 1 , 0, h 1 q 2 + h 2 q 1 + n) G(q 2 , 0, h 1 q 2 + h 2 q 1 + n) = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 , h 1 q 2 2 , l) G(q 2 , h 2 q 2 1 , l) G(q 1 , h 1 q 2 2 , m) × G(q 2 , h 2 q 2 1 , m) G(q 1 , 0, h 1 q 2 + n) G(q 2 , 0, h 2 q 1 + n) . (27)
On the other hand ( 1), ( 7), ( 14), (26) and Lemma 1 imply λ * q 1 , lq 2 q 1 , mq 2 q 1 , nq 2 q 1 λ * q 2 , lq 1 q 2 , mq 1 q 2 , nq

1 q 2 = 1 q 1 q 2 1≤x 1 ,y 1 ,z 1 ≤q 1 e lq 2 q 1 x 1 + mq 2 q 1 y 1 + nq 2 q 1 z 1 q 1 1≤h 1 ≤q 1 e h 1 (x 2 1 + y 2 1 + z 1 + 1) q 1 × 1≤x 2 ,y 2 ,z 2 ≤q 2 e lq 1 q 2 x 2 + mq 1 q 2 y 2 + nq 1 q 2 z 2 q 2 1≤h 2 ≤q 2 e h 2 (x 2 2 + y 2 2 + z 2 + 1) q 2 = 1 q 1 q 2 1≤h 1 ≤q 1 1≤h 2 ≤q 2 e h 1 q 2 + h 2 q 1 q 1 q 2 G(q 1 , h 1 , lq 2 q 1 ) G(q 1 , h 1 , mq 2 q 1 ) G(q 1 , 0, h 1 + nq 2 q 1 ) × G(q 2 , h 2 , lq 1 q 2 ) G(q 2 , h 2 , mq 1 q 2 ) G(q 2 , 0, h 2 + nq 1 q 2 ) . ( 28
)
Using the substitution x → q 2 q 1 x and working as in Lemma 4 we get

G(q 1 , h 1 q 2 2 , l) = G(q 1 , h 1 , lq 2 q 1 ) , (29)
G(q 1 , h 1 q 2 2 , m) = G(q 1 , h 1 , mq 2 q 1 ) , (30)
G(q 2 , h 2 q 2 1 , l) = G(q 2 , h 2 , lq 1 q 2 ) , (31)
G(q 2 , h 2 q 2 1 , m) = G(q 2 , h 2 , mq 1 q 2 ) , (32)
G(q 1 , 0, h 1 q 2 + n) = G(q 1 , 0, h 1 + nq 2 q 1 ) , (33)
G(q 2 , 0,

h 2 q 1 + n) = G(q 2 , 0, h 2 + nq 1 q 2 ) . (34)
Bearing in mind ( 27) -(34) we complete the proof of the lemma.

Lemma 8. Let 8 q. For the function defined by (7) the upper bound λ * (q, l, m, n) q 2 τ 2 (q)

holds. In particular we have λ * (q, l, m, n) q 2+ε .

Proof. Case 1. 2 q. By (1), ( 3), ( 7), ( 14) and Lemma 1 we write

λ * (q, l, m, n) = 1 q 1≤x,y,z≤q e lx + my + nz q 1≤h≤q e h(x 2 + y 2 + z + 1) q = 1 q 1≤z≤q e nz q 1≤h≤q e h(z + 1) q G(q, h, l) G(q, h, m) = 1 q 1≤z≤q e nz q d|q 1≤h≤q (h,q)= q d e h(z + 1) q G(q, h, l) G(q, h, m) = q 1≤z≤q e nz q d|q q d |(l,m) 1 d 2 1≤r≤d (r,d)=1 e r(z + 1) d G(d, r, ldq -1 ) G(d, r, mdq -1 ) = q 1≤z≤q e nz q d|q q d |(l,m) G 2 (d, 1) d 2 1≤r≤d (r,d)=1 e r(z + 1) -(4r) d (l 2 + m 2 )d 2 q -2 ) d = q 1≤z≤q e nz q d|q q d |(l,m) G 2 (d, 1) d 2 K d, z + 1, -4 d (l 2 + m 2 )d 2 q -2 . ( 35
)
Now ( 35), Lemma 1 and Lemma 3 yield

λ * (q, l, m, n) q 1≤z≤q d|q q d |(l,m) τ (d)d -1 2 d, z + 1, 4 d (l 2 + m 2 )d 2 q -2 1 2 q 1≤z≤q d|q q d |(l,m) τ (d) q 2 τ (q)
r|(q,l,m)

1 q 2 τ 2 (q) . ( 36
)
Case 2. q = 2 h q , where 2 q and h ≤ 2. From (36), Lemma 7 and the trivial estimate |λ * (2 h , l, m, n)| ≤ 8 h we deduce |λ * (2 h q , l, m, n)| = λ * 2 h , lq 2 h , mq 2 h , nq 2 h λ * q , l2 h q , m2 h q , n2 h q q 2 τ 2 (q ) q 2 τ 2 (q) . (37)

Now the lemma follows from (36) and (37).

Lemma 9. Assume that 8 q and H 0 ≥ 2. Then for the sums

Λ * 1 = 1≤l≤H 0 |λ * (q, l, 0, 0)| l , Λ * 2 = 1≤n≤H 0 |λ * (q, 0, 0, n)| n , ( 38
) Λ * 3 = 1≤l,m≤H 0 |λ * (q, l, m, 0)| lm , Λ * 4 = 1≤l,n≤H 0 |λ * (q, l, 0, n)| ln ( 39
)
and

Λ * 5 = 1≤l,m,n≤H 0 |λ * (q, l, m, n)| lmn ( 40
)
the estimations

Λ * i q 2+ε H ε 0 , i = 1, 2, 3, 4, 5 hold.
Proof. Using (38) and Lemma 8 we obtain

Λ * 1 , Λ * 2 q 2+ε 1≤l≤H 0 1 l q 2+ε H ε 0
Further (39) and Lemma 8 imply

Λ * 3 , Λ * 4 q 2+ε 1≤l≤H 0 1 l 2 q 2+ε H ε 0
Finally (40) and Lemma 8 give us

Λ * 5 q 2+ε 1≤l≤H 0 1 l 3 q 2+ε H ε 0 Lemma 10.
For any real number ξ and all integers N 1 , N 2 with N 1 < N 2 , we have

N 2 n=N 1 +1 e(ξn) min N 2 -N 1 , ξ -1 .
Proof. See ( [START_REF] Nathanson | Additive Number Theory : The Classical Bases[END_REF], Lemma 4.7).

Proof of Theorem 1

Using (4) and the well-known identity

µ 2 (n) = d 2 |n µ(d) (41)
we get

Γ(H) = 1≤d≤ √ 3H 2 +H+1 µ(d)
where

Γ 1 (H) = 1≤d≤ξ µ(d)Σ(H, d 2 ) , (43)
Γ 2 (H) = ξ<d≤ √ 3H 2 +H+1 µ(d)Σ(H, d 2 ) , (44)
Σ(H,

d 2 ) = 1≤x,y,z≤H x 2 +y 2 +z 2 +z+1≡0 (d 2 ) 1 , (45)
√ H ≤ ξ ≤ H . (46)

Estimation of Γ 1 (H)

In this subsection we follow the method in Chen and Wang [2]. At the estimation of Γ 1 (H) we will suppose that q = d 2 , where d is square-free and

d ≤ ξ. Denote Ω(H, q, x) = 1≤h≤H h≡x (q) 1 . (47)
Now ( 14) and (47) lead to

Ω(H, q, x) = 1 q 1≤h≤H q t=1 e (h -x)t q = 1 q q t=1 e - xt q 1≤h≤H e ht q = H q + 1 q q-1 t=1 e - xt q 1≤h≤H e ht q . (48)
By ( 45), ( 47) and (48) it follows Σ(H, q) = 1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) Ω(H, q, x) Ω(H, q, y) Ω(H, q, z)

= 1 q 3
1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q)

H 3 + 2H 2 S 1 + H 2 S 1 + 2HS 2 + HS 2 + S 3 , (49)
where

S 1 := S 1 (x, q, H) = q-1 t=1 e - xt q 1≤h≤H e ht q , (50)
S 1 := S 1 (z, q, H) = q-1 t=1 e - zt q 1≤h≤H e ht q , (51)
S 2 := S 2 (x, z, q, H) = q-1 t 1 =1 q-1 t 2 =1 e - xt 1 + zt 2 q 2 i=1 1≤h i ≤H e h i t i q , (52)
S 2 := S 2 (x, y, q, H) = q-1 t 1 =1 q-1 t 2 =1 e - xt 1 + yt 2 q 2 i=1 1≤h i ≤H e h i t i q , (53)
S 3 := S 3 (x, y, z, q, H) = q-1 t 1 =1 q-1 t 2 =1 q-1 t 3 =1 e - xt 1 + yt 2 + zt 3 q 3 i=1 1≤h i ≤H e h i t i q . (54)
Taking into account ( 8), ( 49) -( 54) we derive

Σ(H, q) = 1 q 3 H 3 λ(q) + 2H 2 Σ 1 + H 2 Σ 1 + 2HΣ 2 + Σ 2 + Σ 3 , (55)
where

Σ 1 = 1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) q-1 t=1 e - xt q 1≤h≤H e ht q , (56)
Σ 1 = 1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) q-1 t=1 e - zt q 1≤h≤H e ht q , (57)
Σ 2 = 1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) q-1 t 1 =1 q-1 t 2 =1 e - xt 1 + zt 2 q 2 i=1 1≤h i ≤H e h i t i q , (58)
Σ 2 =
1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) q-1

t 1 =1 q-1 t 2 =1 e - xt 1 + yt 2 q 2 i=1 1≤h i ≤H e h i t i q , (59)
Σ 3 = 1≤x,y,z≤q x 2 +y 2 +z 2 +z+1≡0 (q) q-1 t 1 =1 q-1 t 2 =1 q-1 t 3 =1 e - xt 1 + yt 2 + zt 3 q 3 i=1 1≤h i ≤H e h i t i q . ( 60
)
From ( 6), (56), Lemma 6 and Lemma 10 we obtain

Σ 1 = q-1 t=1
λ(q, -t, 0, 0) 1≤h≤H e ht q q-1 t=1 |λ(q, -t, 0, 0)| t q

-1 q q-1 t=1 |λ(q, -t, 0, 0)| t q 2+ε . (61)
Arguing in the same way for the sums (57) -(60) we deduce

Σ 1 q 2+ε , Σ 2 q 3+ε , Σ 2 q 3+ε , Σ 3 q 4+ε . (62)
Now ( 55), ( 61) and (62) imply

Σ(H, q) = H 3 q 3 λ(q) + O H 2 q ε-1 + Hq ε + q 1+ε . ( 63
)
Bearing in mind ( 43), ( 46) and (63) we get

Γ 1 (H) = H 3 d≤ξ µ(d)λ(d 2 ) d 6 + O d≤ξ H 2 d ε-2 + Hd ε + d 2+ε = σH 3 -H 3 d>ξ µ(d)λ(d 2 ) d 6 + O H 2 + ξ 3+ε , (64)
where

σ = ∞ d=1 µ(d)λ(d 2 ) d 6 . ( 65
)
Using Lemma 5 we find

d>ξ µ(d)λ(d 2 ) d 6 d>ξ d 4+ε d 6 ξ ε-1 . ( 66
)
Clearly the function µ(d)λ(d 2 ) d 6 is multiplicative with respect to d 2 and the series (65) is absolutely convergent. Thus (65) and the Euler product give us

σ = p 1 - λ(p 2 ) p 6 . (67)
Now (64), (66) and (67) yield

Γ 1 (H) = p 1 - λ(p 2 ) p 6 H 3 + O H 3 ξ ε-1 + H 2 + ξ 3+ε . (68)

Estimation of Γ 2 (H)

From ( 44) and (45) we write

Γ 2 (H) ξ<d≤ √ 3H 2 +H+1 1≤k≤ 3H 2 +H+1 d 2 1≤z≤H 1≤x,y≤H x 2 +y 2 =kd 2 -z 2 -z-1 1 ξ<d≤ √ 3H 2 +H+1 1≤k≤ 3H 2 +H+1 d 2 1≤z≤H H ε ξ<d≤ √ 3H 2 +H+1 H 3+ε d -2 H 3+ε ξ -1 . (69)

The end of the proof of Theorem 1

Summarizing (42), ( 68), (69) and choosing ξ = H 3 4 we establish the asymptotic formula [START_REF] Tolev | On the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1 is squarefree[END_REF]. This completes the proof of Theorem 1.

Proof of Theorem 2

By ( 5) and (41) we derive

Γ * (H) = 1≤d≤ √ 2H 2 +H+1 µ(d)
1≤x,y,z≤H x 2 +y 2 +z+1≡0 (d 2 )

1 = Γ * 1 (H) + Γ * 2 (H) , (70)
where

Γ * 1 (H) = 1≤d≤ξ * µ(d)Σ * (H, d 2 ) , (71)
Γ * 2 (H) = ξ * <d≤ √ 2H 2 +H+1 µ(d)Σ * (H, d 2 ) , (72)
Σ * (H, d 2 ) = 1≤x,y,z≤H x 2 +y 2 +z+1≡0 (d 2 ) 1 , (73)
√ H ≤ ξ * ≤ H . (74)
5.1 Estimation of Γ * 1 (H) Using ( 9), (73) and proceeding as in Subsection 4.1 we obtain

Σ * (H, q) = 1 q 3 H 3 λ * (q) + 2H 2 Σ * 1 + H 2 Σ * 1 + 2HΣ * 2 + H Σ * 2 + Σ * 3 , (75)
where

Σ * 1 = 1≤x,y,z≤q x 2 +y 2 +z+1≡0 (q) q-1 t=1 e - xt q 1≤h≤H e ht q , (76)
Σ

* 1 = 1≤x,y,z≤q x 2 +y 2 +z+1≡0 (q) q-1 t=1 e - zt q 1≤h≤H e ht q , (77)
Σ * 2 = 1≤x,y,z≤q x 2 +y 2 +z+1≡0 (q) q-1 t 1 =1 q-1 t 2 =1 e - xt 1 + zt 2 q 2 i=1 1≤h i ≤H e h i t i q , (78)
Σ * 2 = 1≤x,y,z≤q x 2 +y 2 +z+1≡0 (q) q-1 t 1 =1 q-1 t 2 =1 e - xt 1 + yt 2 q 2 i=1 1≤h i ≤H e h i t i q , (79)
Σ * 3 = 1≤x,y,z≤q x 2 +y 2 +z+1≡0 (q) q-1

t 1 =1 q-1 t 2 =1 q-1 t 3 =1 e - xt 1 + yt 2 + zt 3 q 3 i=1 1≤h i ≤H e h i t i q . (80)
Now ( 7), (76), Lemma 9 and Lemma 10 lead to

Σ * 1 =
q-1 t=1 λ * (q, -t, 0, 0) 1≤h≤H e ht q q-1 t=1

|λ * (q, -t, 0, 0)| t q -1 q q-1 t=1 |λ * (q, -t, 0, 0)| t q 3+ε . (81)

Working in the same way for the sums (77) -(80) we get

Σ * 1 q 3+ε , Σ * 2 q 4+ε , Σ * 2 q 4+ε , Σ * 3 q 5+ε . (82)
Further (75), ( 81) and (82) give us Σ * (H, q) = H 3 q 3 λ * (q) + O H 2 q ε + Hq 1+ε + q 2+ε . (83) H 3+ε (ξ * ) -1 .

(89)

The end of the proof of Theorem 2

Bearing in mind (70), (88), (89) and choosing ξ * = H 1 2 we establish the asymptotic formula [START_REF] Zhou | On the square-free values of the polynomial x 2 + y 2 + z 2 + k[END_REF]. This completes the proof of Theorem 2.

9 4

 9 +ε which improves the reminder term O H

7

 7

1≤x,y,z≤H x 2 +y 2 +z 2 +z+1≡0 (d 2 )1 = Γ 1 (H) + Γ 2 (H) ,(42)


Alexandre Stienon

Arnaud Fazekas

Jean-Yves Buffière

Pascal Daguier

Ferhat Merchi

Alain Vincent

Jean-Yves Buffiere

A new methodology for predicting fatigue properties of bearing steels: from X-ray micro-tomography and ultrasonic measurements to the bearing lives distribution

Keywords: rolling contact fatigue, b ea ring steel, inclusions, stringers, distribution of fatigue lives

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

Introduction

Non-metallic inclusions are known to have a detrimental effect on the fatigue performance of high strength steels, but producing steels with perfect cleanliness is not possible for mass productions at competitive costs. However, international standards, such as ISO 281 [1], do not include explicitly inclusion charac teristics in the methods of calculating the basic rating life. Hence, predicting the influence of inclusions on bearing lives remains a major challenge for steel producers, bearing manufacturers, and machine indus tries. In previous works [2 -4], a physically based model for predicting the distribution of bearing lives was developed on the basis of spherical inclusions distributed randomly in the steel. In fact, due to the elaboration process of bearing steels, real inclusions have complex shapes and are often lined up, thus forming so-called stringers. To improve the relevance of such a physically based model, these configura tions should be taken into account, especially to be able to describe the influence of the formfog process on fatigue properties. This is the topic of the present paper.

For that purpose, a new methodology has been developed. It is based on (i) the determination of the concentrations of detrimental inclusions in the steel by high frequency ultrasonic testing; (ii) the study of the morphologies of single inclusions or stringers by X-ray micro-tomography, whkh now enables one to obtain three-dimensional (3D) images of inclusions, thanks to the high flux beams of X rays provided by synchrotrons of third generation; (iii) the characterization of the mechanical properties, in terms of elas ticity and plasticity, of the inclusions by nano-indentation; and (iv) the random distribution of inclusions and/or stringers in the bearing steel according to the concentrations provided by ultrasonic data and the computation of the resulting fatigue lives by using a physically based model for crack nucleation and crack propagation up to the surface.

The experimental conditions are described in the first part of the paper. In the second part, some experimental results are illustrated, especially for micro-tomography observations. Then, the modeling conditions are described, and the distributions of fatigue lives predicted by the model are compared to the experimental data derived from fatigue tests performed on flat washer machines. FinaJly, the influence of the orientation of stringers on the fatigue lives is studied.

Experimental Conditions

Material

The work has been carried out on various heats of 100Cr6 steel, heat treated according to standard conditions: 20 min austenitization at 850°C followed by isothermal quenching at 220°C and 1 h temper ing at 170°C. The resulting 0.2 % yield strength is 2360 MPa at room temperature. Additional details about the mechanical properties of the studied material can be found in Ref 2. It should be noted that the studied heats were selected with poor cleanliness according to the purpose of this work.

Ultrasonic Te sting

Ultrasonic testing is commonly used to detect and estimate the concentration of non-metallic inclusions in steel. The main advantage of this technique arises from its ability to scan a large volume of material [START_REF] Auclair | Appropriate Te chniques for Internal Cleanliness Assessment[END_REF]. Details about the technique can be found in dedicated literature [START_REF] Birks | Ultrasonic Te sting[END_REF]. In the present work, this technique has been used for the purposes of (i) estimating the inclusion population in the various heats that have been tested in fatigue and (ii) detecting and locating typical inclusions prior to the X-ray micro-tomography observations. An immersion C-scan Epsilon ® device from Metalscan, 7 equipped wEth an 80 MHz focused transducer, has been used. With this equipment, inclusions of 10 µm in diameter located in the focal zone can be detected. However, the spatial resolution of the technique does not enable one to reveal the detailed shape of inclusions. Such a goal can be reached by using X-ray micro-tomography, as described in the following section.

X-Ray Micro-Tomography

X-ray micro-tomography is a 3D imaging technique that is increasingly used in materials science [START_REF] Buffiere | In Situ X-Ray To mography Studies of Microstructural Evolution Combined with 30 Modeling[END_REF][START_REF] Kak | Principles of Computerized To mographic Imaging[END_REF][START_REF] Baruchel | X-Ray To mography in Material Science[END_REF][START_REF] Banhart | Advanced To mographic Methods in Materials Research and Engineering[END_REF]. It allows one to visualize the inner constituents (pores, inclusions, phases, etc.) of a material, provided they exhibit an X-ray absorption coefficient different from that of the matrix. Although several papers dealing with constituent particles in aluminum alloys can be found in the literature (see, for example, Ref 11), to the best of the authors' knowledge, the detailed characterization of inclusions embedded in a high strength steel matrix has not been reported yet.

The examples of observations presented in this work have been obtained on beam line ID 19 at the European Synchrotron Radiation Facility (ESRF). This beam line provides an X-ray beam with high brightness and high spatial coherence, allowing one to perform micro-tomography with a spatial resolution in the micrometer range. Figure 1 shows the experimental micro-tomography set-up available at beam line ID19 [12]. It includes a monochromator that allows one to select photons with a suitable energy, a scintillator converting X-rays into visible light, and a digital camera to record the radiographs obtained for successive angular positions of the specimen mounted on a rotating stage. To observe inclusions with the high spatial resolution required for this work, the size of the samples has to be limited. This limitation arises mainly from two factors. First, the size of the specimen affects the choice of the energy of the photons. As a rule of thumb, a minimum of 10 % of the intensity of the incident beam should remain transmitted throughout the specimen to get good quality images, which plays in favor of an increasing energy of the photons with increasing the size of the specimen. But, in contrast, a lower energy tends to be required to keep a good contrast of absorption between inclusions and matrix as well as a good efficiency of the sc1ntillator used to detect X-rays. For the present investigations, these conflicting requirements were fulfilled by setting the energy of the incoming photons to 35 keV. Second, the image of the specimen must remain within the field of the charge-coupled device detector of the camera. This depends on the size of the specimen, the optical magnification, and the desired voxel size in the reconstructed image. With the optical magnification chosen, the effective voxel size in the reconstructed 30 image is 0.7 X 0.7

x 0.7 µrn3• AJtogether, these conditions inferred a cross section of 0.4 x 0.4 rnm2 for our needle-like specimens. Once reconstructed by the software available at ESRF (classical filtered back projection method), the 30 images have been analyzed using ImageJ 8 and Amira9 softwares. 30 renditions of the inclusions have been obtained by the region growing segmentation method [START_REF] Russ | The Image Processing Handbook[END_REF].

Experimental Results

Inclusion Observations through X-Ray Micro-Tomography

The various types of non-metallic inclusions most often met in bearing steels have been observed suc cessfully by X-ray micro-tomography: isolated inclusions with a globular shape, stringers constituted of lined up inclusions, and continuous elongated inclusions. Examples of such observations are shown in Fig. 2 for a few inclusions. The 30 renditions are presented after image processing, which enables one to make the matrix transparent and visualize onJy the inclusions in a virtual parallelepiped box showing the limits of the volume analyzed around of the inclusion.

Figure 2(a) shows an example of elongated inclusion that appears as a long continuous. ribbon (length> 1 mm), with its main direction almost parallel to the axis of the bar from which the specimen was cut. Its section is approximately constant throughout the ribbon, typically 8 X 20 µm2• Furthermore, the inclusion is not perfectly straight but rather undulating around its main direction. It should be empha sized that due to this undulating character, the continuity of such type of .inclusion could not be easily evidenced by standard two-dimensional micrographic observations. By means of destructive energy dis persive X-ray analysis, such inclusions have been identified as manganese sulfides in the batches of steels investigated in this work.

Figure 2(b) shows an example of a stringer with lined up inclusions. For a given stringer, the size of the constituent inclusions is widely scattered (from a few microns to 70 µm in the case presented in Fig.

2(b)).

The distances between neighboring inclusions are also widely scattered and were found to vary from a few microns to a few tens of microns for the investigated samples. It should be pointed out that such detailed characteristics of the stringers could not be obtained by ultrasonic echography. Besides, in such stringers, destructive analyses have shown that the constitutive inclusions are alumina inclusions and calcium aluminate inclusions, sometimes with some coating of calcium enriched sulfide. (a) Figure 2(c) shows an example of an isolated globular inclusion. It should be mentioned that they often include two conical ends, as the one selected for the figure. The axis going through the two tips of the cones is nearly a revolution axis of the whole defect and is parallel to the rolJing direction of the bar. The gray level for these conical ends, darker than those of the steel matrix and the inclusion core, suggests that they are cavities. This contrast can be used to remove the conical ends by image processing (Fig. 2(d)). Such globular isolated inclusions have been identified either to alumina or calcium aluminate inclusions.

The flexibility of the tomography representation is illustrated in Fig. 3. It shows three different virtual tomography cuts of the inclusion shown in Fig. 2(c) in orthogonal median planes. The inclusion is globally spherical but exhibits some surface irregularities. The aforementioned cones are also visible in Fig. 3(b) and 3(c) (darker zone).

Finally, Fig. 4 shows a comparison between a virtual tomography cut and a classical micrograph on a polished section of the same inclusion. This comparison confirms the ability of tomography to depict all the details of inclusions with a spatial resolution close to that of optical microscopy. As it will be detailed in the section devoted to modeling, the detrimental character of a stringer is linked to the interaction between constitutive inclusions. This interaction is linked to the distance between neigh boring inclusions, which is widely scattered, as mentioned above. Moreover, the size of the interacting inclusions plays also an important role. Hence, the edge-to-edge distance d between two neighboring inclusions has been normalized by the average of the ractii of these inclusions R111 =(R;+R)12 (see Fig. 5).

In order to characterize this parameter, several 3D images of stringer were analyzed to establish statistical data related to the tested heats. The histogram of the parameter di R111 thus obtained is given in Fig. 6. It shows that the maximum of the distribution of the normalized distance occurs for di R1 1 1 values close to one. For further use of these data, it can be seen that the distribution can be reasonably fitted by a log-normal distribution (continuous curve in Fig. 6).

Stringer Concentration Obtained through High Frequency Ultrasonic Te sting

The stringers can be easily detected by using the ultrasonic equipment referenced in the previous section.

For that purpose, parallelepiped samples were taken from the laminated bars, machined, and then scanned with the point focused 80 MHz probe. The scanning plane was parallel to the bar axis. Figure 7 shows an example of a C-scan image thus obtained. As pointed out previously, the resolution of the technique does not enable one to distinguish the details of the stringer. Only its total length L (see Fig. 5) can be estimated. Thus, the ultrasonic testing and the X-ray micro-tomography technique appear quite complementary. Furthermore, for each heat tested in this work, a total volume of about 4 cm3 has been investigated. The stringers have been classified according to their length. An example of histogram of the length of the stringers is shown in Fig. 8. Most of the stringers have a length shorter than 200 µm, but a few were found to belong within categories of longer length. Finally, it should be mentioned that, up to now, the ultrasonic testing was not able to recognize the nature of inclusions in the detected stringers. Hence, the nature of the inclusions, or at least the majority of them, has to he known using another technique.

Computational Modeling of the Distribution of the Fatigue Lives

The approach used for predicting the distribution of fatigue lives in the present work consists in introduc ing the effects of interaction of neighboring inclusions in stringers in a model initially designed for isolated spherical inclusions, which has been presented in previous works [2][START_REF] Lormand | From Cleanliness to Rolling Fatigue Life of Bearings-A New Approach[END_REF][START_REF] Fougeres | A New Physically Based Model for Predicting the Fatigue Life Distribution of Rolling Bearings[END_REF]. This is made possible due to both the internal characteristics of the stringers provided by X-ray micro-tomography and the statistical data provided by ultrasonic testing.

The computational modeling of the bearing lives distribution includes two parts. The first one is of

Determinist Model fo r Crack Nucleation and Growth [2]

The material is loaded cyclically under the repeated contact produced by the rolling bodies. Crack nucleation originating from inclusions arises from the stress concentration around the inclusions. This induces cyclic micro-plasticity leading to the accumulation of dislocations. Finally, crack nucleation oc curs once a critical density of dislocations is reached in this area.

In the present work, the contact stress field is that produced by circular balls (9.992 mm in diameter) on a plane surface. The resulting contact stress, 4.2 GPa, is that of the fatigue tests on flat washers described in the last section of the paper. The stress concentration around the inclusions has been com puted by means of the Abaqus finite element software, as described in Ref 14. Moreover, in this reference, it has been shown that the real geometry of the inclusions provided by the 30 images from micro tomography can be reasonably modeled by simplified shapes in order to derive by finjte element a realistic stress field around the inclusions (for instance, the inclusion shown in Figs. 2(c) and 3 can be valuably modeled by a spherical core associated with two conical empty sides [START_REF] Stienon | A New Method ology Based on X-Ray Micro-Tomography to Estimate Stress Concentrations Around Inclusions in High Strength Steels[END_REF]). The elasto-plastic behavior of the 100Cr6 matrix and the inclusions are also given in this reference. The accumulation of dislocations in the vicinity of inclusions is characterized by the damage accumulation factor f" defined in Ref 2, and the criterion for crack nucleation can also be found in Ref 2. The resulting cycle number for crack nucleation is Nw The orientation of the micro-crack is given by the direction of the maximal shear stress for dislocation glide. Its initial length is taken equal to that of the zone where localized dislocation movements occur, i.e., the stress concentration zone around the inclusion [2].

Crack propagation is calculated by considering a physically based model as in Ref 3. The crack is assumed to grow in mode n with a drcular shape. The amplitude of the stress intensity factor for crack growth is calculated by considering an effective shear su•ess that accounts for friction of the crack lips due to the compression stress normal to the crack plane. To account for the limited extension of the Hertzian loading, the width extension of the crack is bounded to the contact width of the rolling ball. The current crack length, a, is calculated numerically by integrating the local crack growth rate dal dN, which is updated (in relation to the present crack length and the crack position in the loading stress field) after each numerical step. The number of cycles to bearing failure, N P , is considered necessary for the crack to reach the surface. This means that the number of cycles from the surface breaking stage of the crack to complete spalUng is neglected.

Finally the total cycle number to failure, NR, is given by the sum NR = N11+Np.

influence of the interaction between inclusions

In this work, the excess of harmfulness of stringers has been assigned to the interaction between neigh boring inclusions, which reduces the number of cycles to crack nucleation Nw Indeed, the superposition of the stress concentration zones around the neighboring inclusions enhances locally the stress concentration. This mechanism is illustrated in Fig. 9 for a simple stringer made of three spherical inclusions of equal diameter, with the axis of the stringer oriented perpendicularly to the contact surface. Figure 9 shows the Tresca stress maps in a plane passing through the axis of the stringer for different normalized distances di Rm between the inclusions. It appears clearly that the local stress increases markedly when di Rm is decreased down to 0.5. It should be pointed out that any local stress increase generally induces a stronger effect on the local cydic micro-plasticity that govems the damage accumulation. Moreover, it has been shown elsewhere [START_REF] Courbon | Influence of Inclusion Pairs, Clusters and Stringers on the Lower Bound of the Endurance Limit of Bearing Steels[END_REF] that the number of inclusions in the stringer does not influence significantly the maximal value of the local stress. In other words, for a given inclusion the stress concentration in the surrounding matrix is mainly influenced by the next neighboring inclusion. This fact enables one to characterize the interaction between constitutive inclusions by studying simple stringers made of three inclusions. Furthermore, a stress enhancement factor has been defined for the stringers by where:

1.2 --+--90° 1.15 --o--60° 1.1 • 30°..
[ ma x ] K = O" Tr stringer r [ ma x ]
(j Tr single I

[ cr�r a x Jstringer and [O"�r a x Jsingle 1= maximal values of the Tresca stress in the steel matrix in presence of interaction between neighboring inclusions and for a single inclusion of the same type, respectively.

The stress enhancement factor, Kn for alumina stringers has been plotted versus the normalized distance between neighboring inclusions in Fig. 10 for various orientations of the stringer with respect to the contact surface. As already mentioned, the stress enhancement increases steeply as the normalized distance decreases. Moreover, Fig. 10 shows that Kr is also markedly influenced by the orientation of the stringer, the orientation perpendicular ito the contact surface (orientation of 90°) being the more detrimen tal. It is worth mentioning that this stress enhancement factor has to be calculated prior to the statistical calculations described in the following paragraph. Thus, the stress concentrations do not need to be computed again for every particular stringer, but they can be simply derived from a single constitutive inclusion. This method drastically reduces the computation time for the statistical predictions.

Statistical Prediction of the Distribution of Fatigue Lives

The statistical implementation of the model for predicting the distribution of fatigue lives has been carried out in the following way.

•

For each considered steel heat, a series of virtual flat washers is built using a Monte Carlo method for drawing successively the number of stringers of a given type in each flat washer, the position of each stringer center (uniform djstribution), its total length, and the corresponding di R111 parameter. It should be specified that the number and the total length of each stringer are drawn according to the experimental distribution determined by ultrasonic echography (see example in Fig. 8), while the di R,,, parameter is drawn according to the distribution determined by X-ray micro-tomography (see Fig. 6).

• Every flat washer is subjected to a virtual fatigue test, and the number of cycles N F to fatigue failure is computed by applying the models for crack nucleation and growth described above for every stringer in every flat washer; according to the weakest link rule, the fatigue life N F of a given washer is defined as the minimum value among all the values corresponding to the various string ers; once they are analyzed, all these minimum values form the distribution of lives related both to the heat and the conditions of fatigue testing.

••+ •••• "

""""""""""""

+ Fatigue tests •••X --Simulated 5% ••+ --Simulated 95% 1.E+04 1.E+06 HeatD + fatigue tests ••• X •• Simulated 5% •• +•• Simulated 95% 1.E+02 1.E+04 1.E+06 x • HeatE 0. + • + 1.E+02 x x+ ><• + 1.E+02 -• •• +••••••••••••••----• + Fatigue tests • • •X • • Simulated 5% • • + • • Simulated 95% 1.E+04 1.E+06 0 HeatG 0 ... • +•••-•••--•••••••••• • .+ .• + Fatigue tests --• X--Simulated 5% ••+ •• Simulated 95% 1.E+04 1

Application and Comparison with Experimental Data

The assessment of the methodology for predicting the distribution of fatigue lives has to be carried out by comparing the computed fatigue lives to experimental ones. This has been achieved by investigating eight different heats chosen with different cleanliness levels. For that purpose, on the one hand, rolling contact fatigue tests have been carried out on flat washer specimens using SNR Roulements type FB2 test rigs [START_REF] Baudry | Fatigue Properties Evaluation of Bearing Steel Produced by Rotary Continuous Casting[END_REF]. On the other hand, the cleanliness of every heat has been characterized by ultrasonic testing, as explained previously (see Fig. 8). The experimental tests were conducted according to the "sudden death testing method" [START_REF] Baudry | Fatigue Properties Evaluation of Bearing Steel Produced by Rotary Continuous Casting[END_REF]. Thus, 50 washers were tested by groups of five units under a nominal Hertzian pressure of 4.2 GPa. For time reasons, fatigue tests without failure after running JOOO h or so were interrupted.

On the simulation side, for a given heat, first a series of 10 OOO virtual washers has been built up and the corresponding lives have been calculated. From experience feedback, the constitutive inclusions in the distributed stringers were assumed to be alumina inclusions in the presented computations. This procedure provides the overall rolling fatigue properties of the heat, which can be presented as the cumulative probability of failure versus life, in a classical logarithmic diagram. Figure 11 shows an example of such a representation for heat D characterized in Fig. 8. As shown in earlier works [START_REF] Lormand | From Cleanliness to Rolling Fatigue Life of Bearings-A New Approach[END_REF], the overall cumulative distribution of fatigue lives exhibits a typical f shape. The tendency of the cumulated failures to saturate well below 100 % is linked with the significant probability to have washers with 111 0 detrimental stringers in the core of the Hertzian stress field.

Finally, in order to compare simulations with experimental data, the simulated data have to be ana lyzed according to the "sudden death" method. Moreover, since a single experimental test exhibits a random character, the scattering range of the results has to be determined. For that purpose, 100 virtual sudden death tests of 50 washers were built up from the 10 OOO virtual tests. The cumulative distribution function of each number of failure (Nos. 1-5) can be derived from these results. Then, the scattering range of the results with a confidence interval of 90 % was obtained from these distribution functions. Finally, the five experimental data points and the computed scattering range were plotted in the same Weibull diagram for every heat. Four examples are reported in Fig. 12.

First, it can be seen that the computed scattering ranges are generally wide. As a matter of fact, in Fig. 12 the scattering range appears to be narrower for heat D that is less clean than the other heats. Such a relationship between the scattering width and the inclusion concentration can be explained easily. For a clean heat the probability to find a harmful inclusion in the stressed region below the raceway is low, and hence fatigue failure is a quite random process. In contrast, for a less clean heat the probability to find harmful inclusions in the same region is close to one, thus reducing the randomness of the fatigue process. Second, the experimental data fall well within the computed scattering ranges, except for some interrupted tests (open symbols in Fig. 12). In fact, these results remain consistent with the predicted ranges because an interrupted test potentially refers to a real life longer than the number of cycles to interruption. A similar agreement was obtained for all the investigated heats. All these good agreements show that the new model presented in this section is able to predict reasonably well the statistical distribution of fatigue lives of bearings in relation with the steel cleanliness characterized by hjgh frequency ultrasonic testing and X-ray micro-tomography. It is worth pointing out that estimating the scattering ranges by means of the experi mental fatigue tests would take about 4 years for one heat with seven rigs running continuously. This shows the great advantage of the modeling route.

Finally, regarding the influence of the stringer orientations, Fig. 13 shows the comparison between the distributions of fatigue lives for stringers oriented parallel and perpendicular to the contact surface. It appears that for the perpendicular orientation (90°), the first failures occur significantly earlier than for the parallel orientation (0°). This is mainly due to the higher stress concentration for the perpendicular orien tation, which has been shown in Fig. 10. In fact, such a larger stress significantly reduces the number of cycles to crack nucleation, but it does not influence the tendency to saturate as much. This is not surprising because, as mentioned above, the saturation level of cumulated failures is mainly linked to the concentra tion of stringers, which was the same for both simulations. It should be mentioned that this influence of the orientation is in good agreement with the trend observed experimentally, which exhibits a L10 fatigue life about five times longer for washers machined from a bar to obtain the "parallel orientation" of the stringers than for those machined from the same bar to obtain the "perpendicular one." Unfortunately, for these results, the high frequency ultrasonic characterization of the heat was not available, and hence, the corre sponding simulation was not possible.

Conclusion

A new methodology has been developed for predicting the distribution of fatigue lives of bearings from appropriate characteristics of the inclusion population in the steel.

This work has shown that high frequency ultrasonic testing provides reliable data on concentrations of harmful stringers in bearing steels. X-ray micro-tomography has been used for the first time to visualize the 3D shape of inclusions and stringers. This new technique appears to be quite complementary of the high frequency ultrasonic testing. Indeed, it is able to provide the internal details of stringers, but it is limited to small volumes, while in contrast ultrasonic testing enables one to investigate larger volumes of material but without distinguishing internal details of stringers.

The physically based model presented here makes a clear link between the cleanliness of the steel and its fatigue performances. It has been validated by comparing the predicted distribution of lives wiith a set of fatigue test data carried out according to the sudden death method.

Moreover, this modlel is able to (i) account for the influence of the stringers orientation with respect to the contact surface, (ii) make predictions of distribution of fatigue lives for large batches of bearings representative of populations of bearings used in industry, and (iii) estimate the scattering of results for fatigue tests performed on reduced numbers of parts.

Furthermore, it should be pointed out that, in its present form, the applicability of the model is limited to operating conditions for which inclusion initiated damage is the controlling process for bearings failure. For instance, this excludes applications with extremely high contact pressures involving bulk metal fatigue or with very clean steels and surfaces indented by pollution of the lubricant.

In the future, this methodology could be used to optimize the forming process of bearing steels. For instance, the reduction ratio of bars or tubes used for manufacturing bearings governs the characteristics of inclusions and particularly of stringers. Using the presented methodology would enable one to predict the fatigue properties of the final product versus the parameters of the forming process.

A cknowledgments

The ASCOMETAL company is gratefully acknowledged for its continuous support . The writers aEso wish to thank Dr. G. Baudry and Dr. E. Henault for fruitful discussions and encouragements.

[1] ISO 281 , 2007, "Rolling Bearings-Dynamic Load Ratings and Rating Life," International Standards Organization, Geneva, Switzerland.

[2] Vincent, A., Lormand, G., Lamagnere, P., Gosset, L., Girodin, D., Dudragne, G., and Fougeres, R.,

  FIG. I-Experimental set-up available at line ID19 for micro-tomography experiments.

8

  1mageJ: http://rsb.info.nih.gov/ij/.

FIG. 2 -

 2 FIG. 2-3D renditions of typical inclusions observed by X-ray micro-tomography in bearing steels: (a) Non-fragmented elongated inclusion; (b) fragmented elongated inclusion; (c) globular inclusion with adjacent conical ends; and (d) same inclusion as (c) but the conical ends have been removed by image processing.

FIG. 3 -

 3 FIG. 3-Reconstructed tomography slices of the globular inclusion shown in Fig. 2(c) observed in three perpendicular median planes. y axis is parallel to the rolling axis of the bar. Two cones appear in darker gray in (b) and (c) on each side of the inclusion. The dotted line in picture (c) corresponds to the observation plane in Fig.4.

FIG. 4 -FIG. 5 -

 45 FIG. 4-Comparison between (a) X-ray micro-tomography and (b) optical micrography of the globular inclusion shown in Figs. 2(c) and 3.

  FIG.6-Histogram of Jhe normalized distances between inclusions in the stringers (for the normalized distance axis, numbers refe r to the medium value of each class whose width is I); the corresponding distribution is fitted by a log-normal function (continuous curve).

100µmFIG. 7 FIG. 8 -

 78 FIG.7-C-scan image of a stringer detected by high frequency ultrasonic echography.

FIG. 9 -

 9 FIG. 9-Tresca stress maps for a stringer oriented perpendicularly to the contact surface fo r various distances between constitutive inclusions: (a) di R111 =8; (b) di R111 =2; (c) di R111 =0.5; case of alumina inclusions; and applied stress taken at the center of .the Hertz field fo r a Hertz pressure of 4.2 GPa.

.FIG. 10 -

 10 FIG.10-Stress enhancement factor, Kn for alumina stringers as a function of the normalized distance between neighboring inclusions for various orientations of the stringer (0 ° =parallel to the contact surface; 90° =perpendicular to the contact surface).

FIG. 11 -

 11 FIG. 11-Simulated cumulative probability of failure versus Life for JO OOO flat washers plotted in a log-log diagram for heat D; 1 h =675 OOO cycles.

  FIG. 12-Comparison between the experimental cumulative probability of fa ilure versus life for sudden death tests plotted in a We ibull diagram and the computed scattering ranges for a confidence interval of 90 o/o: Examples of heats labeled C, D, E, and G. The open symbols indicate that the corresponding tests were interrupted.

FIG. 13 -

 13 FIG.13-Comparison between the predicted cumulative probability of fa ilure versus life for stringers oriented parallel or perpendicular to the contact suiface (inclusion concentrations are those of heat D ).

Metalscan, Industrial Echography: www.metalscan.fr.

Amira: www.amira.com.