J.-P Allouche

Joseph Kung

The zeta-regularized product of odious numbers

What is the product of all odious integers, i.e., of all integers whose binary expansion contains an odd number of 1's? Or more precisely, how to define a product of these integers which is not infinite, but still has a "reasonable" definition? We will answer this question by proving that this product is equal to π 1/4 2ϕe -γ , where γ and ϕ are respectively the Euler-Mascheroni and the Flajolet-Martin constants.

Introduction

Extending or generalizing "simple" notions is a basic activity in mathematics. This involves trying to give some sense to an a priori meaningless formula, like √ -1, 1/0, n≥1 n, etc. Among these attempts is the question of "assigning a reasonable value" to an infinite product of increasing positive real numbers. This question arises for example when trying to define "determinants" for operators on infinite-dimensional vector spaces. One possible approach is to define "zetaregularization" (see the definition in Section 2 below). The literature on the subject is vast, going from theoretical aspects to explicit computations in mathematics but also in physics (see, e.g., [START_REF] Hawking | Zeta function regularization of path integrals in curved spacetime[END_REF][START_REF] Spreafico | Finite temperature quantum field theory on noncompact domains and application to delta interactions[END_REF]): we will -of course-not give a complete view of the existing references, but rather restrict to a few ones to allude to general contexts where these infinite products take place. Our purpose is modest: to give the value of an infinite arithmetic product (namely the product of all odious integers, i.e., of all those integers whose binary expansion contains an odd number of 1's, see, e.g., [START_REF]On-Line Encyclopedia of Integer Sequences[END_REF])

1 × 2 × 4 × 7 × 8 × 11 × 13 × 14 × 16 × 19 × 21 × 22 × 25 × 26 × 28 × 31 × 32 × . . .

Definitions. First properties. Examples

Definitions

The remark that log n i=1

λ i = - d ds n i=1 1 (λ i ) s s=0 1
suggests a way of defining the infinite product of a sequence (λ i ) i≥1 of positive numbers (see, e.g., [START_REF] Quine | Zeta regularized products[END_REF][START_REF] Soulé | The determinant of Laplace operators[END_REF]) by means of zeta-regularization: suppose that the Dirichlet series ζ Λ (s) := i 1/λ s i converges when the real part of s is large enough, that it has a meromorphic continuation to the whole complex plane, and that it has no pole at 0, then the zeta-regularized product of the λ i 's is defined by

∞ i=1 λ i := e -ζ ′ Λ (0)
(this definition clearly coincides with the usual product when the sequence of λ i 's is finite). If ζ Λ has as pole at 0, there is a slight generalization of the definition above (see [START_REF] Illies | Regularized products and determinants[END_REF][START_REF] Kurokawa | Generalized zeta regularizations, quantum class number formulas, and Appell's O-functions[END_REF], also see [START_REF] Kimoto | Some examples of generalized zeta regularized products[END_REF]):

∞ i=1 λ i := e -Res s=0 ζ Λ (s) s 2
where Res s=0 g(s) stands for the residue at 0 of the function g. To put some general deep context about this definition, in particular about infinite determinants, the reader can consult [START_REF] Deninger | On the Γ-factors attached to motives[END_REF][START_REF] Manin | Lectures on zeta functions and motives (according to Deninger and Kurokawa)[END_REF][START_REF] Voros | Zeta Functions over Zeros of Zeta Functions[END_REF].

First properties

The following equalities hold * For all N ≥ 1,

∞ i=1 λ i = N i=1 λ i ∞ i=N +1 λ i . * For a > 0, ∞ i=1 (aλ i ) = ∞ i=1 λ i a ζ λ (0) * If A and B form a partition of the positive integers, then ∞ i=1 λ i = i∈A λ i i∈B λ i .

Examples

One can find several examples in the literature, (taken, e.g., from [START_REF] Kimoto | Some examples of generalized zeta regularized products[END_REF][START_REF] Kurokawa | A generalization of Lerch's formula[END_REF][START_REF] Kurokawa | Generalized zeta regularizations, quantum class number formulas, and Appell's O-functions[END_REF][START_REF] Quine | Zeta regularized products[END_REF][START_REF] Yoshimoto | Analytic Number Theory (Beijing/Kyoto[END_REF] or deduced from properties in Section 2.2):

n≥0 (n + x) = √ 2π Γ(x) (for x > 0; Lerch formula [ * ]) n≥1 n = √ 2π (Lerch formula for x = 1), hence, e.g., n≥1 (2n)
= √ π, and

n≥1 (2n + 1) = √ 2 n≥0 (n 2 + 1) = e π -e -π (general Lerch formula [ * ]) n≥0 (n 2 -n + 1) = e π √ 3 2 + e -π √ 3 2 [ * ] n≥0 (n 4 + 1) = 2 cosh(π √ 2) -cos(π √ 2) [ * ] n≥1 n n = e -ζ ′ (-1) = Ae -1/12 (A = 1.2824271 . . . is the Glaisher-Kinkelin constant [ * * ]) n≥0 a n = a -1/
  n≥0 1 (n + x) s   s=0 = log Γ(x) √ 2π • Actually Lerch proved (cited in [15, Equality (2), p. 942]) the general formula n≥0 ((n + x) 2 + y 2 ) = 2π Γ(x + iy)Γ(x -iy)
which of course implies the classical Lerch formula. This general Lerch formula was generalized in [START_REF] Kurokawa | A generalization of Lerch's formula[END_REF] where ((n + x) 2 + y 2 ) is replaced with ((n + x) m + y m ).

[ * * ] Recall that the Glaisher-Kinkelin constant A = 1.2824271 . . . can be defined in several ways (see, e.g., [START_REF] Van Gorder | Glaisher-type products over the primes[END_REF] and the references therein):

A = lim n→∞ 1 1 × 2 2 × 3 3 × ... × n n n n 2 +n/2+1/12 e n 2 /4 = e 1 12 -ζ ′ (-1) = (2π) 1 12 e γ 12 -ζ ′ (2) 2π 2 =   2πe γ p prime p 1 p 2 -1   1/12
.

[***] As indicated in [START_REF] Kurokawa | Generalized zeta regularizations, quantum class number formulas, and Appell's O-functions[END_REF] one may recall that formally

n≥1 n = ζ(-1) = -1/12.
Another example of zeta-regularized product is given by the Fibonacci numbers (F n ) n≥1 in [START_REF] Kitson | The regularized product of the Fibonacci numbers[END_REF] where ∞ n=1 F n is computed in terms of the Fibonacci factorial constant and the golden ratio or in terms of the derivative of the Jacobi theta function of the first kind and the golden ratio. Of course the result needs the study of the Dirichlet series 1/F s n : other similar Dirichlet series or zeta-regularized products are studied in [START_REF] Behera | On Lucas-balancing zeta function[END_REF] and [START_REF] Dutta | Regularized products over balancing and Lucasbalancing numbers[END_REF].

Up to generalizing the notion of zeta-regularized product ("super-regularization"), one has ( [START_REF] García | The product over all primes is 4π 2[END_REF], also see [START_REF] Smirnov | The ζ-regularized product over all primes[END_REF]):

p prime p = 4π 2 .
In what follows, we let O denote the set of odious positive integers, i.e., of positive integers whose sum of binary digits is odd, and E the set of evil numbers, i.e., of positive integers whose sum of binary digits is even. We let a(n) denote the characteristic function of odious numbers (i.e., a(n) = 1 if the sum of binary digits of n is odd, and a(n) = 0 if it is even) and ε n = (-1) a(n) (note that ε n = 1 -2a(n)). Clearly ε n is equal to +1 if the sum of the binary digits of n is even, and to -1 if the sum is odd; in other words (ε n ) n≥1 is (up to its first term) the famous Thue-Morse sequence on the alphabet {-1, +1} (see, e.g., [START_REF] Allouche | The ubiquitous Prouhet-Thue-Morse sequence[END_REF]).

Theorem 3.1 We have with the notation above, with

Q = n≥1 2n 2n + 1 εn , n∈O = (2π) 1/4 Q -1/2 n∈E = (2π) 1/4 Q 1/2 Also Q = 2 -1/2 e γ ϕ
where ϕ is the Flajolet-Martin constant [START_REF] Flajolet | Probabilistic counting algorithms for data base applications[END_REF].

Proof. For ℜs > 1 we have

ζ O (s) = n∈O 1 n s = n≥1 a(n) n s = n≥1 1 -ε n 2n s = 1 2 ζ(s) - 1 2 g(s)
where ζ is the Riemann zeta function and g(s

) := n≥1 ε n n s •
But, by [2, Theorem 1.2] with q = 2 (also see [9, Lemma 1]) g can be analytically continued to the whole complex plane, and it satisfies, for all s ∈ C,

g(s) = -1 + k≥1 (-1) k+1 s + k -1 k g(s + k) 2 s+k • (1)
This implies that g(0) = -1 and

g(s) -g(0) s = k≥1 (-1) k+1 (s + 1)(s + 2) . . . (s + k -1) k! g(s + k) 2 s+k
hence, by letting s tend to 0:

g ′ (0) = k≥1 (-1) k+1 g(k) k2 k •
On the other hand, mimicking a computation in [2, p. 534], one has

k≥1 (-1) k+1 g(k) k2 k = k≥1 (-1) k+1 k2 k n≥1 ε n n k = n≥1 ε n k≥1 (-1) k+1 k2 k n k = n≥1 ε n log 1 + 1 2n = -log Q where Q := n≥1 2n 2n + 1 εn . So that g ′ (0) = -log Q. We thus obtain ζ ′ O (0) = 1 2 ζ ′ (0) - 1 2 g ′ (0) = - 1 4 log(2π) + 1 2 log Q which finally yields n∈O n = e 1 4 log(2π)-1 2 log Q = (2π) 1/4 Q -1/2 . Now n∈O n n∈E n = n≥1 n = √ 2π which gives n∈E n = (2π) 1/4 Q 1/2 .
It remains to recall that the Flajolet-Martin constant ϕ [9] is equal to

ϕ := 2 -1/2 e γ 2 3 n≥1
(4n + 1)(4n + 2) 4n(4n + 3) εn = 0.77351 . . . and thus ([8, Section 6.8.1], also see [START_REF] Allouche | Thue, combinatorics on words, and conjectures inspired by the Thue-Morse sequence[END_REF])

ϕ := 2 -1/2 e γ Q hence Q = 2 -1/2 e γ ϕ • Remark
ζ O S (s) = n∈O 1 (n + 1) s = n≥1 a(n) (n + 1) s = 1 2 n≥1 1 -ε n (n + 1) s = 1 2 n≥0 1 -ε n (n + 1) s = 1 2 ζ(s) - 1 2 f (s)
where f (s) = n≥0 ε n (n + 1) s . It was proved in [START_REF] Allouche | Dirichlet series and curious infinite products[END_REF] that this function f can be analytically continued to the whole complex plane, and that f

′ (0) = log 2 2 . Hence ζ ′ O S (0) = -1 4 (log(2π) + log 2) = -1 4 log 4π. This gives finally n∈O S n = 2 1/2 π 1/4 and hence n∈E S n = π 1/4 .

Beyond odious and evil

The main tool used in the computation of the zeta-regularized product of odious or of evil numbers above is the "infinite functional equation" [START_REF] Allouche | Thue, combinatorics on words, and conjectures inspired by the Thue-Morse sequence[END_REF]. A multidimensional analog of this equation exists for "automatic" sequences [START_REF] Allouche | Automatic Dirichlet series[END_REF]. We could expect a similar result in the general case of a zetaregularized product of integers having an automatic characteristic function. What makes things more complicated in the general case is that the involved zeta function occurs as a component of a vector of Dirichlet series satisfying an infinite functional equation, but that the zeta function itself does not necessarily satisfy such an equation. Furthermore this vector is meromorphic but not necessarily holomorphic on C (in particular 0 might be a pole). As suggested by the referee, looking at the subcase given by the parity of block-counting sequences could well be the right generalization of the Thue-Morse case. Trying to follow this suggestion, we only arrived at a partial result for, e.g., the Golay-Shapiro (also called Rudin-Shapiro) sequence where, instead of considering the parity of the number of 1's in the binary expansions of integers for the Thue-Morse sequence, one considers the number of 11's in the binary expansions of integers. We hope to revisit these questions in the near future.

3 . 2

 32 Instead of considering the odious and evil numbers, one might have considered -in a rather non-natural way-the shifted odious and evil numbers, namely the sets O S := {n+1, n ∈ O} and E S := {n + 1, n ∈ E}. Then n∈O S n = exp(-ζ ′ O S (0)). But, with the notation of the proof of Theorem 3.1, and using that ε 0 = 1,

Acknowledgments We thank M. Marcus for having detected misprints in the first version of this paper. We also thank J.-Y. Yao, B. Saffari, and the referee for their constructive remarks.


E Bellec

M L Facchinetti

C Doudard

S Calloch

S Moyne

M P Silvestri

Modelling and Identification of Fatigue Load Spectra: Application in the Automotive Industry

Keywords: Load spectrum, automotive chassis system, analytical model, parameter identification 1

This paper is focused on variable amplitude loading spectra applied in the automotive industry, more specifically for the chassis system parts with respect to high cycle fatigue design. A first analytical model referred to as Heuler's model is considered. It is of the simplest use, with one parameter to be identified. A whole identification process, based on road measurement realized on carmaker's proving ground with instrumented vehicles, is developed and discussed. Once the model is identified, the cycles influence on damage production is also investigated. In the light of the obtained results with this model, a second one, noted modified Heuler's model, is introduced and investigated. This model, similar to the first, requires the identification of two parameters. Once again, the whole model identification process is applied, and the lifetime accuracy is assessed. A new tool is presented, enabling the visualisation on the loading spectrum representation of the most and less damaging cycles.

Hence, a methodology is set to legitimize a gate implementation when studying a variable amplitude loading spectrum.

Introduction

This paper is an extended version of that reported in [START_REF] Facchinetti | Fatigue Load Spectra: Model and identification proposals for the automotive[END_REF]. The use of Variable Amplitude Loading (VAL) spectra for structure life assessment has met a growing interest among numerous fields since the end of the 20 th century ( [START_REF] Sonsino | Why Variable Amplitude Loading ? A Key for Lightweight-Structural Durability Design[END_REF], [START_REF] Costa | Fatigue behaviour of AA6082 friction stir welds under variable loadings[END_REF], [START_REF] Correia | Fatigue crack growth modelling of Fão Bridge puddle iron under variable amplitude loading[END_REF], [START_REF] Vantadori | A novel procedure for damage evaluation of fillet-welded joints[END_REF]). As outlined in [START_REF] Sonsino | Why Variable Amplitude Loading ? A Key for Lightweight-Structural Durability Design[END_REF], these spectra can be defined from representative load-time measurements (one flight sequence for an overseas passenger aircraft, one-year operation of an offshore platform) or from pre-defined standards (sequences of specific cornering or deceleration for a passenger car). Numerous recommendations exist in the literature ( [START_REF] Sonsino | Why Variable Amplitude Loading ? A Key for Lightweight-Structural Durability Design[END_REF], [START_REF] Vantadori | A novel procedure for damage evaluation of fillet-welded joints[END_REF], [START_REF] Johannesson | Guide to load analysis for durability in vehicle engineering[END_REF], [START_REF] Köhler | Load Assumption for Fatigue Design of structures and Components[END_REF], [START_REF] Facchinetti | Fatigue damage of materials and structures assessed by Wöhler and Gassner frameworks: recent insights about load spectra for the automotive[END_REF], [START_REF] Facchinetti | Load Spectra and fatigue damage: applications to the automotive industry[END_REF]) to set up a circumstance-relevant load case spectrum. Once the representative spectrum is defined, it is applied several times during experiments or numerical simulations until a structure failure appears (e.g. from incipient material crack to definitive integrity loss).

Taking a closer look to the automotive industry, carmakers do not share a standard, nor homologation protocol when it comes to fatigue design. Every firm is free to draft its own fatigue assessment methodology using its own specific proving grounds and relevant load cases. The imperative is to ensure the vehicle security, parts integrity and durability and every carmaker must implement a reliable self-certification process. Regarding the fast-changing market, technical cooperation is now an asset. Hence, it is of general interest to share and compare different existing frameworks.

As highlighted in Table 1, many field-specific variable load-time histories have been determined since the 1970's. Through their publications ( [START_REF] Heuler | Generation and use of standardised load spectra and load-time histories[END_REF], [START_REF] Heuler | Standardised load-time histories -A contribution to durability issues under spectrum loading[END_REF], [START_REF] Sonsino | Fatigue testing under variable amplitude loading[END_REF], [START_REF] Berger | Betriebsfestigkeit in Germany -an overview[END_REF]), P. Heuler & al. describe the various spectrum design steps respecting every physical specificity. As far as the automotive industry is involved, since the 1990's, at least two established variable loading spectra families draw attention regarding fatigue design. The first one, named CAR Loading Standard (CARLOS), gathers reference spectra regarding loads (uniaxial and multiaxial considerations) on suspension parts, powertrain components and car trailer coupling. These spectra are mainly used in the German carmaking industry as fatigue design references. The second proposal is named STAndardized Multiaxial Simulation (STAMAS) ( [START_REF] Grubisic | Determination of load spectra for design and testing[END_REF], [START_REF] Heim | Early stage rig testing durability approval[END_REF]). The vehicle life situations are classified regarding the road condition, and the manoeuvers performed (braking, cornering). This approach has been recently applied to the fatigue strength assessment of chassis system components of a city car [START_REF] Voigt | Fatigue strength assessment for components and subsystems of a lightweight, space saving city car with electric drive[END_REF], nevertheless it is not of wide knowledge, nor application.

These two spectra families are standardised time-loading sequences. To fast forward the fatigue design process and ease the comparison of spectra, VAL spectra should be easily customized depending on the vehicle, the road condition, the driving situation. This study is focused on an analytical spectrum model, based on Road Load Data (RLD) measured both from vehicles operating on the manufacturer proving ground and by real customers on daily usage. Both material and structure design of automotive safety parts vary accordingly to the vehicle of interest. As not to confine oneself to studying a specific safety part, a choice is made in this article to design an analytical model based on the loads measured at the entry point of the chassis system, i.e. the wheel.

The scope of this study is high cycle fatigue design ( [START_REF] Bathias | La Fatigue des matériaux et des structures -Fatigue of materials and structures[END_REF], [START_REF] Suresh | Fatigue of Materials[END_REF]) (i.e. without any yield phenomena) under uniaxial load. The load spectrum design process is illustrated in Figure 1. Regarding measurements performed on the manufacturer proving-ground, the vehicle wheels are equipped with standard force transducers with a 1000 Hz data-acquisition frequency. The measured spectrum usually arises from the concatenation of several vehicle specific manoeuvers and road conditions from the manufacturer proving ground. The sequence of operations is defined by the manufacturer, based on customer feedback ( [2], [START_REF] Vantadori | A novel procedure for damage evaluation of fillet-welded joints[END_REF]). Each manoeuver is multiplied by a specific number of times to obtain a representative loading of an ordinary vehicle usage. The spectrum detailed manoeuver composition remains at the manufacturer discretion. The identified model is deemed reliable if its induced damage is equivalent to the one generated by the RLD measurement [START_REF] Raoult | Fatigue equivalent load approach for fatigue design of uncertain structures[END_REF]. 1) No of load components or additional variables (time, temperature etc.); 2) approx. 3) partly the references do not cite the original report(s), but those available to the public

Figure 1: Load spectrum design steps and model reliability criterion

The first part of this article highlights the approaches found in the literature to represent a structure life assessment under variable amplitude loading (section 2.1). Loadings studied in this article are measured at the wheel. Even during the most critical manoeuvers, in the framework of normal usage, the maximum stress undergone is expected far from the vehicle parts yield stresses.

The RLD spectrum and its damage are represented (section 2.2). Even if very low amplitude cycles are not expected to cause relevant fatigue damage, no cycle omission is applied a priori. Only basic temporal noise filtering is considered. No endurance limit is considered. The damage calculation, specific to this study, is based on a

Basquin-like formulation [START_REF] Basquin | The exponential law of endurance tests[END_REF]. Hence, the impact of the Basquin slope value is investigated (section 2.3) and an analytical tool named "iso-damaging curve" is used to detect the most impacting amplitude range from a given loading spectrum (section 2.4).

Variable amplitude framework

Usually, fatigue life assessment is represented by means of Wöhler curve [START_REF] Wöhler | Versuche zur Ermittlung der auf die Eisenbahnwagenachsen einwirkenden Kräfte und die Widerstandsfähigkeit des Wagen-Achsen[END_REF] (Figure 2) combining, for a given material (and process), the number of cycles until a given failure criterion (e.g. crack initiation), " ", to the loading amplitude " ". This curve is built up thanks to experiments and is expected to be applied for constant amplitude loadings.

Both parameters "b" (>1), the Basquin slope and "B" (>>1), the Basquin constant, are material dependent to be identified. It has been reported through literature reviews [START_REF] Bergamo | Préconisations pour les caractéristiques statistiques de résistance en fatigue -Applications aux aciers et autres matériaux utilisés dans la construction automobiles[END_REF], that " " does not only vary accordingly to the material but also to the process, i.e. the assembly procedure. For instance, the Basquin slope, "b" is expected to range from 2 to characterize a welded joint (minimum found for any application), to 8 and even more for a steel plate. For automotive applications, please consider that if the Basquin slope is assumed to be bigger than 8, high cycle fatigue requirements are not expected to be relevant anymore. Considering "B" fixed, the more "b" increases, the more higher loading amplitudes are damage producing.

As the Wöhler curve is applied for constant amplitude loading, in 1939, E. Gassner [START_REF] Gassner | Festigkeitsversuche mit wiederholter Beanspruchung im Flugzeugbau -Strength tests under repeated loading for aeronautical engineering[END_REF] presented a new approach to characterize structure lifetime assessment under VAL. The main innovation is not to represent the loading spectrum thanks to an S-n curve but replacing actual cycles "$ " (% representing the amplitude range) by cumulative occurrences noted "& " (Figure 3). They are calculated using the following relationship

& ∑ $ ( ()* . (2)
The maximum amplitude is set to " * ". At this amplitude the number of cycles is noted "$ * & * ". At the spectrum lowest amplitude, the spectrum overall length is noted " & + ". The first loading spectrum designed by Gassner was built for the aeronautic industry. It is inspired from a gaussian-like curve divided into 8 amplitude blocks. Its simplicity was imposed by the experimental limitations set at that time.

Nowadays, much more complex and realistic loading spectra can be reproduced on test benches or computed by numerical simulations. Once the maximal amplitude " ", the overall length " & + ", and the shape are defined, the spectrum can be applied several times to the structure until failure. Hence, the lifetime can be assessed thanks to cumulative occurrences, regarding the number of times the overall loading spectrum has been applied. To be accepted as an elementary spectrum, its overall length should be one order of magnitude under the lifetime expected. E. Gassner suggested a Wöhler-like curve to represent structures lifetime under a given variable loading spectrum. Both curves can be represented thanks to the number of cycles up to failure " ". The main difference lies in the amplitude dependency. The regular Wöhler curve is depicted thanks to the constant amplitude loading " ". Once the shape of the spectrum is set, the Gassner one is " " dependent. Thus, the overall lifetime has been reported to the ",-.

-,-. " curve. The failure curve equation is another Basquin-like formula noted ( 0.

(3)

Once again, the ("k", "K") are material/process parameters to be identified. Previous researches show the equality between "1" and " " values from the Gassner and the Basquin formula. This has been mathematically

proven if the Palmgren-Miner damage accumulation law is applied: "k = b". Actually, as both slopes' values are supposed to represent the same material physical behaviour, it is of common sense to consider them equal.

Figure 3: Standard high cycle fatigue frameworks, Wöhler and Gassner curves

Comparing both failure curves on Figure 3, the Gassner one seems to quantify a longer lifetime than the Wöhler one. Even if the slope value is the same ("b = k"), the value of "K" and "B" are different. The impression of longer lifetime is a graphical mirage. Both curves are not represented in the same frame even if they are similar:

("S", "N") for the Wöhler curve and (" ", "N") for the Gassner one. To shift from one representation to the other, P.Heuler (6,7) defined the following coefficient, named Spectrum Shape Factor (SSF),

2 log 6 ∑ 7 8
∑ 7 8 69 8 /9 ;<= > ? >.

(4)

It essentially depends on the spectrum shape and the Basquin slope. If the VAL turns out to be a constant one, equal to " ", the SSF would be equal to 0 and both curves would match (reference frames end up the same one). This factor is always definite positive as "b" and " / " are respectively greater and lower than 1. The factor is lower than the unit value (<1) for spectra with many large and few small cycles. On the contrary, its value rises when the number of cycles at the highest amplitudes is preeminent compared to the lowest ones. Heuler An SSF value may correspond to several loading spectra. Hence, knowing only the maximum loading amplitude and the spectrum corresponding SSF value is not enough to depict all the spectrum loading complexity.

It is therefore required to use the entire loading spectrum to identify a reliable loading model for fatigue design.

Spectrum representation and damage-like tool formulation

Before using any of the fatigue design frameworks introduced previously, the measured spectrum coming from RLD is displayed in Figure 4. (A choice is made by the authors regarding the spectrum display reference frame: ("S-n") or ("S-H"). The reader must keep in mind that the RLD introduced is representative of a car loading experienced in real life situation.

Figure 4: VAL coming from RLD, represented via effective cycles (a) and cumulative occurrences (b)

Representing the loading spectra thanks to the loading amplitude ratio " / " eases the comparison between different RLD patterns e.g. impact of the vehicle tested, the executed manoeuvers, the road conditions … Regarding the abscissa, the first graph, Figure 4 a) turns out to be quite irregular and thus tough to be described via an analytical model. On the contrary, the second graph, representing the same loading spectrum as the first one but using cumulative occurrences, is smoother and eases the identification process. The other advantage of cumulative occurrences representation is the independency towards the amplitude scale size.

As expected, there are much more cycles at the lowest amplitudes than at the highest ones. The spectrum identification from 0.5 to 0.1 seems to be easily achievable. Nevertheless, there are some step-like irregularities at the highest amplitudes that might be difficult to represent. Thus, both frameworks have some shortcomings in order to seek for a simple analytical model to fit the overall spectrum shape.

As these loading spectra are meant to design a chassis system under high cycle fatigue, it is proposed to compare the damage induced by the RLD and the one coming from the identified model to decide which analytical loading model is best fitting. The damage calculation formula used in this study is not based on the cumulative occurrences ("H") but on the effective number of cycles ("n") per amplitude. By using the Palmgren-Miner's rule ( [START_REF] Palmgren | Die Lebensdauer von Kugellagern -The Fatigue Life of Ball-Bearings[END_REF], [START_REF] Miner | Cumulative damage in fatigue[END_REF]), the overall damage induced by the spectrum, "B CDC ", is then given by

B CDC ∑ E ∑ $ / ∑ $ " /#. (5)
For each amplitude range " ", the damage "E " is equal to the number of cycles executed "$ ", divided by the lifetime at the same amplitude " ". Assuming that the Basquin formula is used to assess the structure lifetime under constant amplitude loading, " " can be replaced by "#/ " ". The "B" value is a material parameter to define. In the study instance, the loading spectrum is measured at the vehicle wheel. Thus, the damage calculation is not based on any specific chassis part material. To ease the comparison between RLDs performed on different vehicles, "B" is fixed to " FGH " ", (this normalization helps to notice if the damage spectrum shape differs significantly). The damage equation per amplitude block is noted

E $ 6 / > " . (6)
This formula has no proper physical meaning. It just enables to link the amplitude ratio to the damage and compare the different damage spectra coming from several measurements together. Based on this damage-like formula with a Basquin slope value of 5, the damage induced by the RLD presented in Figure 4 is calculated.

Figure 5: Damage (a) and cumulative damage (b) induced by RLD (b=5)

The Figure 5 highlights the impact of the highest amplitudes on the lifetime (from 0.8 to 1 for " / ").

The Figure 5 b) represents the cumulative damage, "B IJ ", according to the spectrum loading amplitude ratio. The cumulative damage is increasing with the loading ratio to finally reach the overall damage "B CDC " at the highest amplitude " ".

This damage representation illustrates some useful information. The maximum amplitude loading block (refer to Figure 5 b) upper right corner) induces almost 10% of the overall damage with only 0.01% of the overall spectrum length (referring to the number of cycles in the spectrum). The other way around, at the lowest amplitude, 10% of the overall damage is induced by 98% of the spectrum overall length, corresponding to an amplitude ratio from 0 up to 0.35. Definitively, working with cumulative damage ("B IJ 0" for " 0" and "B IJ B CDC "

for " ") instead of regular damage is the opportunity to eliminate a majority of the fluctuations, the amplitude scale definition dependency, and to analyze at a glance the cumulative damage curve slope and the final value, respectively.

Material parameters impact on the damage spectrum shape

This chapter reports the damage spectrum shape, based on the damage calculation formula, for several Basquin slope values. As the Basquin constant "# " " is "b" dependent, it is senseless to compare the cumulative damage curves one with another without damage normalization. The cumulative damage curves shape should be compared assuming each curve is normalized respectively by its overall damage induced. The Figure 6 highlights three normalized cumulative damage curves, all calculated from the RLD, with different Basquin slope values "b".

Loading amplitude ratio, S/S max Loading amplitude ratio, S/S max

Figure 6: Normalized cumulative damage induced by RLD, with different Basquin slope values

The higher is b, the highest amplitudes impact on damage calculation is more significant. For information, regarding these specific loading sequences:

• "b = 2". 98% of the spectrum length (Number of cycles $ from " / 0 to 0.4") induces 65% of the overall damage at the lowest amplitude.

• "b = 5". 98% of the spectrum length induces 20% of the overall damage. 0.01% of the spectrum length, belonging to the highest amplitude induces almost 10% of the overall damage.

• "b = 8". 98% of the spectrum length induces 3% of the overall damage. 0.01% of the spectrum length, belonging to the highest amplitude induces 30% of the overall damage.

Considering the future model identification process, this aside reveals that the more "b" increases, the more the model accuracy should be focused on higher amplitudes. If there is no contrary specification, a Basquin slope value "b = 5" (according to automotive literature for average quality components) is considered in the rest of this study.

Amplitude of interest using iso-damaging tool

The previous section enables to find the most damaging amplitude (depending on the Basquin slope value) by looking at the cumulative damage graph. Considering the studied loading spectrum and regardless of the " " value, all the amplitudes do not participate to the same extent in the damage. This section aims to detect the most damaging amplitudes, directly from the loading spectrum representation using a pragmatic tool: "the isodamaging" curves.

Iso-damaging curves are supposed to cause the same amount of damage per amplitude range, i.e. a uniform damage through the entire spectra. Hence, they are " " dependent. The iso-damaging curves presented in this section are also calculated using the damage formula (6) presented in section 2.2. Once the overall amount of damage induced "B CDC " and the number of loading amplitude classes "N" are set, the iso-damaging curve can be derived. The damage for the loading amplitude class " " can be related to "B CDC " and "N" by

∀ % P⟦1, N⟧ , E B CDC /N $ 6 / > " . (7)
This equation can be reformulated to determine the number of cycles for each loading amplitude class, "$ ", for a

given "B CDC ",

$ 6B CDC /N> * 6 / > V" . (8)
The lower the cumulative damage curve slope gets (definite positive), the more the corresponding amplitude range is damaging. When the amount of damage per amplitude range is defined, the corresponding number of cycles can be determined. Hence, the corresponding loading spectrum can be displayed (Figure 7). For this study, the size of the amplitude classes is fixed to 50N. (Order of magnitude: 1%, compared to the " " measured)

Figure 7: Iso-damaging curves: cumulative damage framework (a) and associated loading spectra (b)

In this study, the iso-damaging curves are used as a tool to compare the damages induced for different amplitude classes from the RLD spectrum. When the RLD curve matches an iso-damaging curve at a certain point, it indicates that the damage induced at this specific amplitude range by the loading spectrum is equal to the one caused by the iso-damaging curve.

Figure 8: Comparison between the RLD and the iso-damaging curves

Considering the Figure 8, three main observations can be expressed:

-At the highest amplitude, the RLD is as damaging as the "B CDC 10 X " curve.

-At the medium amplitudes, the most damaging amplitude ranges are around " / 0.4" regarding the RLD spectrum. Still, the loading spectrum induces less damage at these amplitudes than the "B CDC 10 X " curve.

-At the lowest amplitudes, " / < 0.2", the loading spectrum is less damaging than "B CDC 10 \ ".

It is even lower than "B CDC 10 ] " when " / < 0.1".

Once the damage formula is set, using the iso-damaging tool is of great interest to compare the most and least damaging cycles for a given spectrum shape. These observations highlight that cycles at the lowest amplitudes, even if they compose at least 50% of the overall spectrum length, are 100 times less damaging than cycles at the highest amplitudes. To go further, this tool may be used to legitimize the omission of some nondamaging cycles from the initial spectrum, also known as "gate implementation" process. The gate amplitude corresponds to the amplitude under which no cycles are actually counted into the spectrum.

Heuler's model identification

This section aims to present the VAL model formerly proposed by P. Heuler in the literature ( [START_REF] Heuler | Generation and use of standardised load spectra and load-time histories[END_REF], [START_REF] Heuler | Standardised load-time histories -A contribution to durability issues under spectrum loading[END_REF]).

A first theoretical study is provided regarding the different spectra shapes that can be modelled (section 3.1). Then, the model identified based on the RLD previously introduced and its fatigue life assessment accuracy is evaluated (section 3.2).

Loading amplitude ratio, S/S max

Model formulation and theoretical investigation

The analytical model formula designed by P. Heuler ( [10], [START_REF] Heuler | Standardised load-time histories -A contribution to durability issues under spectrum loading[END_REF]) is log6&> /log 6& + > 1 -6 / > ^.

(9)

This model seems quite flexible as P. Heuler used it to clarify and compare all the spectra loading shapes considered by Gassner. It is defined based on cumulative occurrences and loading amplitude ratio. The coefficient "_" is named shape exponent. This one parameter model dependent (considering the parameters responsible for the scaling "& + " and " " are already known) can reproduce several different VAL. The Figure 9 illustrates different spectra shapes according to "_", with "& + 10 `" cycles. As the cumulative occurrences' framework can be tricky to figure out, the graph b) illustrates the same spectra regarding the effective number of cycles. The model can reach a wide variety of behaviours especially at the lowest amplitudes.

Figure 9: Heuler's model load spectra, via cumulative occurrences (a) and regular number of cycles (b)

Three main spectra behaviours emerge from this model regarding the shape exponent value:

• "ν < 1": much lower number of cycles at the highest amplitudes than at the lowest. "&" is increasing faster while the amplitude is decreasing. These spectra would be considered for soften driving behaviour.

• "ν = 1": linear relationship between the amplitude ratio and the cumulative occurrences ratio.

• "ν > 1": spectra slope is lower at the highest amplitudes than at the lowest. The more the coefficient increases, the more the maximum effective number of cycles is reached at a higher amplitude. These spectra would feat a stiffer driving behaviour.

Loading amplitude ratio, S/S max Loading amplitude ratio, S/S max Table 3, coming from [START_REF] Heuler | Generation and use of standardised load spectra and load-time histories[END_REF], correlates some of the spectra's shape depending on ν to the SSF value introduced in section 2.1. Based on P. Heuler's previous work, the shape factor coefficient is expected to sway between 0.5 and 5

to match the various standardized variable load-time histories presented in Table 1.

In the scope of this study, this spectrum model aims to be used for fatigue design. Hence, in the same vein of section 2.2, the damage spectra coming from Heuler's model is investigated. As the shape factor impacts the overall damage value, only the normalized cumulative damage curves are inspected in Figure 10. Once again, the value "ν = 1" sets the distinction between the different damage spectra shapes. Regarding the "ν = 0.5" curve, the damage cumulative slope is bigger than the other ones. This implies that less damage is provided per amplitude class. It is worth noting that 50% of the overall damage is caused/produced before reaching 40% of the maximum amplitude. The other curves slopes ("ν ≥ 1") are quite similar between 10% and 90% of the Loading amplitude ratio, S/S max overall damage. The main differences between the curves occur before 10% and after 90% of the maximum loading amplitude.

The illustrated damage curves in Figure 10 correspond to a reference Basquin slope ("b = 5").

Accordingly, to the section 2.3, the Figure 11 illustrates the model cumulative damage curve dependence to " ".

The higher "b", the less significant in damage calculation are low amplitudes, the more the cumulative damage slope of "ν = 0.5" grows (Figure 11b)). On the contrary, if "b" gets closer down to 2, the cumulative damage slope value of "ν = 0.5" is getting closer to the other slopes as the lowest amplitudes gain impact significance.

Model spectrum identification and damage accuracy assessment

As stated previously, Heuler's model is identified on the loading spectrum coming from RLD illustrated in Figure 4. The identification is set in the (,-.6&>/log 6& + > -/ ) frame using standard least square identification methods to fit the model to the data. The identified spectrum overall length is fixed to the initial & + (~ 10 `> coming from the measurement (Figure 12). Heuler's model main characteristic is the fixed starting point, no matter the shape factor value, at the coordinates (" * / 1; ,-.6& * >/,-.6& + > 0"), which means to have one cycle at" ". In order to fit the loading spectrum, the model tries to catch up this starting level and the following several "steps", from higher amplitudes towards the lower ones.

The comparison graphs (Figure 12b)) of both cumulative damage curves highlights the impact of the amplitude range between 30% and 60% of the maximum amplitude ("b = 5": 55% of the RLD damage is made between these two amplitude ratios). Even if there is a slight gap between the two loading curves below " / = 0.6", the model inaccuracy gets higher as the frame is a ratio based on logarithmic scale. This identification is not accurate enough as the shape obtained (" c. dc") illustrates a much stiffer behaviour than the initial one.

The induced damage is ultimately 3 times bigger than the initial one. Thus, if, experiments were carried out with the identified spectrum instead of the initial RLD, the lifetime would be assessed 3 times shorter, in terms of number of cycles, than the expected one.

The way the RLD spectrum is built (concatenation of several specific manoeuvers) deeply impacts its shape: several cycles are found at the highest amplitudes. This article aims to find a model that may fit in the scope of fatigue design for all the RLD types used by the carmakers. Hence, a model modification must be applied to adapt the spectrum shape specifically at the highest amplitudes and let free the starting loading point value (" * / 1; & * ").

Loading amplitude ratio, S/S max Loading amplitude ratio, S/S max Modified Heuler's model identification

Model formulation and theoretical investigation

To provide some flexibility considering the depicted behaviours at the highest amplitudes, another parameter, noted "g", is added to the initial Heuler's model. The relationship (9) becomes log6&> /log 6& + > 1 -g6 / > ^.

(10)

Given the model mathematical construction, when " * ", then "& * & + 6*Vh> ", when "& * " was fixed to 1 with the previous model. Physically, "& * " is bounded between 1 and "& + ". Hence, "g" is expected between 0 and 1. The previous model corresponds to "g 1.". The Figure 13 illustrates the impact of "g", considering different value of "_". Once again, the loading spectra are represented using cumulative occurrences and effective cycles, respectively.

Figure 13: Modified Heuler's model load spectra, via cumulative occurrences (a) and number of cycles (b)

The shape factor "_" from the initial Heuler's model is kept in this model. Hence, stiff ("_ 2.5") and smooth ("_ 0.5") behaviours can still be represented. The "g" parameter influences the loading spectra at the highest amplitudes only. The more "g" decreases, the more the loading curves are shifted to the right. No matter the "g" value, the variety of behaviours at the lowest amplitudes induced by "_" are quite the same of the initial model curve.

Now that the loading spectra curves are set up, the Figure 14 highlight the cumulative damage caused by these spectra. As the "g" value also affects the overall damage value, only the cumulative damage curves shapes are studied. Each one is respectively normalized by its overall damage induced. Each graph corresponds to one fixed "_" value and illustrate the impact of "g" on cumulative damage curve shape. The more "g" decreases, the more the damage is high amplitudes sensitive. The effect tends to be the same caused by the Basquin slope increase (section 2.3). This is due to the start level reached when " * ".

As " * & * & + 6*Vh> ", any "g" reduction induces a growing shift.

Model spectrum identification and damage accuracy assessment

In the same way as described in section 3.2, the modified Heuler's model parameters are identified from the RLD. Both "g" and "_" parameters are left free for the identification. The least squares method is once again applied. The Figure 15 highlights the model identification result. To consider the starting level at the highest amplitudes, the model set its parameter "g" to 0.59 ("log6& * > /log 6& + > 0.4"). Then, even if there are some visible "steps" on the RLD until the amplitude ratio 0.7, the model is drawn as an almost straight line, ("_ 1.04") and practically matches the smoother curve part.

The behaviour depicted is then less stiff than the one identified with the first model ("_ 2.32"). Considering both model identifications (Figure 12 and Figure 15), it is noticeable that the modified Heuler's model damaging curve steadily follows the initial one. Even if this model smooths the starting upper "stairs", the lifetime assessment is very close. Thus, the modified Heuler's model fits at least the requirements regarding fatigue design for the specific loading spectra found in the automotive industry.

Gate implementation

In the section 2.4 a specific tool, named "iso-damaging" curves, has been introduced to detect simultaneously, from the RLD considered, both the most and the less damaging amplitudes. The idea investigated

Loading amplitude ratio, S/S max Loading amplitude ratio, S/S max in this paragraph is to remove from the RLD the least damaging cycles. Then, the modified Heuler's model is once again identified but on the truncated model. The Figure 16 illustrates the Gate implementation on the RLD.

Figure 16: Gate implementation illustration on RLD spectrum

First, the maximum iso-damaging curve is identified between the amplitude ratios " / 0.7" and " / 0.1". This curve corresponds to the iso-damaging spectrum that induces the most damage and matches the RLD curve in at least one point. The 0.9 limit is set because of the curve stair shape that might bias the identification. The most damaging amplitude corresponds to the amplitude ratio " / 0.4".

To implement the gate, the cycles from the RLD that induce less damage than a third of the maximum iso-damaging curve identified can be cut-off from the overall spectrum. This choice is not arbitrary, previous considerations toward this kind of RLD highlight that at least 98% of the overall damage is kept using this cutting off ratio.

The modified Heuler's model is once again identified from the "truncated" RLD. Both parameters are then set to "g 0.57 "and "_ 1.66" instead of "g 0.59 "and "_ 1.04" for the overall model. It is tricky to compare both identification as the spectrum overall length has changed due to the gate implementation. Still, as expected, both "g" values are very similar indeed. As illustrated in Figure 17 the gate implementation slightly influences the behavior identified, but not the fatigue lifetime accuracy.

Loading amplitude ratio, S/S max %t-B IJC %t-B 3

Conclusion

This paper highlights a VAL analytical model identification process considering chassis system for high cycle fatigue design. The study is limited to the uniaxial RLD measured directly at the car wheels. These data are basic inputs for the overall chassis system design, no matter of the material, process, technology and geometry diversity of an automotive chassis system.

At first, the spectra analysis using cumulative occurrences representation is shown to offer several advantages. In this framework, an analytical model is proposed and investigated: the modified Heuler's model, inspired by the literature, is based on two parameters whose identification makes effective use of the cumulative representation formulation. To be endorsed, the identified model must cause the same overall damage as the initial RLD, but also corresponds to a certain extent to the damage induced per amplitude range, to keep the strongest physical meaning. The use of cumulative damage curve gathers both information. The model's new "g" parameter enables the identified spectrum to consider a non-unitary cycle spectrum at the highest amplitude. The model's old "_" parameter keep on transcribing a wide variety of driving behaviours, especially at the lowest amplitudes.

Thus, within the fatigue assumptions made the modified Heuler's model is suitable to effectively represent field RLD spectra.

Moreover, further investigations on the damage impact per amplitude class reveals that regardless of how much cycles belong to the lowest amplitudes, they are either not or few damaging. The study of the Basquin slope " " value significance and the use of "iso-damaging" curves helped to develop an effective gate implementation methodology. Given an RLD example set with " 5", 99% of the overall damage induced by the initial spectrum is conserved by removing 87% of the cycles at the lowest amplitudes. And the identified model on the truncated data does not deeply vary from the initial identification.

This study has two main prospects: the development of analytical models designed for the multiaxial framework and the modified Heuler's model identification on standardized spectra coming from other fields (e.g. aeronautics, offshore, …) to compare the spectra shapes and set new standards.

6

Figure 2 :

 2 Figure 2: Standard high cycle fatigue framework for constant amplitude loading, Wöhler curve

  investigated the SSF coefficient based on Gassner's first VAL spectrum. The SSF value is about 2.5. The Gassner curve would be shifted from the Wöhler curve at least of two orders of magnitude on the right. Heuler investigated number of cycles to failure amplitude SSF also several VAL spectra developed in the industry and found the SSF coefficient, reported in Table 2 (using "b 5" as the Basquin slope value).

Figure 10 :

 10 Figure 10: Normalized cumulative damage curves, Heuler's model illustration (b = 5)

Figure 11 :

 11 Figure 11: Normalized cumulative damage curves, Heuler's model illustration b = 2 (a), b = 8 (b)

Figure 12 :

 12 Figure 12: Heuler's model identification on RLD (a), c. dc, and cumulative damage curves (b)

Figure 14 :

 14 Figure 14: Normalized cumulative damage depending on " " (b =5), (a). i; (b) ; (c) c. i

Figure 15 :

 15 Figure 15: Modified Heuler's model identification on RLD (a), α = 0.59, . , and cumulative damage curves (b)

Figure 15 b

 15 Figure 15 b) illustrates the damage cumulative curve caused by the modified model compared to the one coming from the RLD. Comparing both graphs, it is noticeable that the slight shapes difference at the amplitude 0.4 in the loading spectra representation induces the slight cumulative damages curves separation. Still, the overall damage values are quite close ("B IJ klm /B IJ Dnop ~0.9"). Approximately, the same lifetimes are obtained between the two spectra (10% difference on the induced damage indicator). This remains acceptable in terms of lifetime assessment.

Figure 17 :

 17 Figure 17: Truncated RLD and model identification ( . iu), ( . ), loading spectra (a) and cumulative damage curves (b)

  Modified Heuler's model, on Truncated RLD Truncated RLD Modified Heuler's model, on RLD RLD

Table 1 : European standardized sequences and load-time histories ([10])

 1

	Name	Purpose		Structure detail		Description of load history	Load	Block size	Equiv.	No. of load	Year/
							channels 1)	(Cycles) 2)		usage	level	ref 3)
		Car	loading	Vertical (ve), lateral	Random with occasional fluctuations of	1	ve: 136,000	40,000 km ≤ 64	1990/
	CARLOS	uniaxial standard	(3	forces (la), longitude (lo), on front	= -0.18 (ve), -0.64 (la), -1.6 (lo) mean stress, mixture of 5 road types, R		lo: 84,000 la: 95,200			[20]
		sequences)		suspension parts
	CARLOS multi	Car (multiaxial) loading standard	4-channel suspension parts components for front load	Time histories, sample frequency 0.005 components based on guide functions sec, correlation functions between load	4	Similar uniaxial CARLOS	to	40,000km	/	1994/ [21]
		General purpose	Narrow-band to wide-	I=0.99 -no mean stress fluctuation.	1	1.0*10 6		-	20	1974/
	GAUSSIAN	random sequence	band random (3 levels	I= 0.70-minor mean str. fluctuation.		1.4*10 6			[22]
				of irregularity, I)		I= 0.33-considerable mean str. fluct.		3.3*10 6
		Transport aircraft	Wing root bending	Constant positive mean stress. For gust	1	402,000		4,000	20	1973/
	TWIST	wing root		moment		loads, ground-air-ground cycles (GAG)				flights	[23]
						= underloads
	WAWESTA	Stell mill drive	Drive components	train	Sequence of 10.000 milling runs	1	28,200		1 month	23	1990/ [24]

Comparison of both damage spectra Measured loading spectrum

	Nomenclature	Road Load Data: Signal based on several specific vehicle manoeuvers
	Load time signal Notation list Loading amplitude vector	The whole signal is supposed to cover the vehicle lifetime expectation
	Loading amplitude numbered i. The amplitudes are classified in descending order
	Identified loading spectrum
	Data treatment 1 Maximum amplitude of the loading spectrum	Model reliability assessment:
	Number of cycles to failure (valid for constant and variable amplitude loadings)
	Number of cycles to failure corresponding to the constant amplitude loading
	Cumulative occurrences vector
	Cumulative occurrences, from	to ,
	Cumulative occurrences, spectrum overall length
	Number of cycles performed per amplitude block for a given loading spectrum
	Number of cycles performed at a given amplitude range
	Basquin slope, material parameter
	Basquin constant, material parameter
	Spectrum shape factor
	Gassner slope, material parameter (= , Basquin parameter)
	is finally proposed to find and remove from spectra either the least or non-damaging cycles (section 4.3).
	Damage vector
	Damage induced per amplitude range, for a given loading spectrum
	1 : The rainflow counting method is used all along this study to count the number of cycles per loading amplitude
	range. Other counting method exist such as range-pair, level-crossing, … Total damage induced by the overall spectrum
	Cumulative damage vector
	Model shape exponent parameter (applied in Heuler's model and modified Heuler's model)
	Model starting point parameter (applied in modified Heuler's model)

The theoretical assumptions and damage calculation methods are depicted (section 2.2). Throughout the article, considering the fatigue assumptions made, the most impacting amplitudes are investigated (sections 2.3 & 2.4). The RLD measurement is set as a reference. To represent the whole RLD complexity and find a standard model, the attention is drawn to an analytical loading spectrum formula (section 3). The lifetime model estimate accuracy is then discussed. Regarding the achieved results, a slight modification to this formula is suggested. Once again, the model theoretical behaviour and damageinduced accuracy is investigated (sections 4.1 & 4.2). A methodology, based on theoretical fatigue-related tools,

Representation and damage assessment of recorded load spectra This

  section aims to set the study theoretical framework. The usual methodologies applied to assess fatigue lifetime are depicted, at first under constant loading, then under variable amplitude loading (section 2.1).

Table 2 : European standardized sequences corresponding SSF values (extracted from [10])

 2

	Name	SSF
	CARLOS	2.66
	CARLOS multi	2.70
	GAUSSIAN	2.32
	TWIST	3.09
	WAWESTA	1.97

Table 3 : Spectrum shape parameters ν and SSF for typified amplitude spectra (extracted from [10])

 3

	Spectrum		SSF	Description
	1	∞	0	Constant amplitude loading
	2	4	1.46	ν > 2 typical for bridge and crane structures
	3	2	2.45	Stationary gaussian random process
	4	1	3.77	Typical for road roughness induced loads
	5	0.8	4.21	ν ≤ 1 typical for wind gusts, wave actions, etc.

Acknowledgements

This work was carried out within the framework of the partnership between Groupe PSA and the OpenLab Computational Mechanics with the financial support of the ANRT for the CIFRE contract n°2019/0764.


Yumna Qureshi
email: yumna.qureshi@ensta-bretagne.org

Mostapha Tarfaoui
email: mostapha.tarfaoui@ensta-bretagne.fr

Khalid Lafdi

Electro-thermal-mechanical Performance of a sensor based on PAN Carbon Fibers and Real-Time Detection of Change Under Thermal and Mechanical Stimuli

Keywords: Real-time monitoring system, PAN carbon fiber sensor, electromechanical performance, electro-thermal behavior

Structural health monitoring (SHM) is a vastly growing field consisting of sensors embedded in or attached with the structure which respond to the strain or other stimuli to monitor the deformation in real-time. In this study, a carbon fiber (CF) sensor was developed using unidirectional Polyacrylonitrile (PAN) carbon filaments aligned straightly together and its sensitivity was calculated experimentally, with the gauge factor (GF) in 10.2-10.8 range. The electro-thermal behavior of this CF sensor showed distinct performance and detected the change in the surrounding temperature. There is a good reproducibility in the results in both piezoresistive and electro-thermal behavior of the CF sensor and its electrical performance showed real-time detection of both mechanical and thermal stimuli. The results established that the CF exhibited good potential as a flexible strain sensor for in-situ monitoring of damage or energy release during the failure of composites.

Introduction

The detection of local damage such as delamination, interlaminar failure, matrix softening, and matrix cracking in composites is often difficult to detect unless the performance of the materials has been compromised [START_REF] Simon | Smart fabrics technology development[END_REF]- [START_REF] Sassi | In-situ heat dissipation monitoring in adhesively bonded composite joints under dynamic compression loading using SHPB[END_REF]. Non-destructive techniques (NDT) such as ultrasonic detection, X-rays, etc. can detect local damage however they often require disassembly of the structure for inspection and they aren't able to detect damage in instantaneously. Acoustic emission is often used for real-time monitoring of the failure in structures but, interpretation of the data is a complex process and mostly qualitative. So, it is important to develop novel techniques to monitor the deformation of the structure in real-time, and structural health monitoring (SHM) is a renowned and extensively used system to study the behavior of the structure in real-time to guarantee their reliability and safety [START_REF] Qureshi | Development of microscale flexible nylon/Ag strain sensor wire for real-time monitoring and damage detection in composite structures subjected to three-point bend test[END_REF]- [START_REF] Qureshi | In-situ Monitoring, Identification and Quantification of Strain Deformation in Composites under Cyclic Flexural Loading using Nylon/Ag Fiber Sensor[END_REF].

Currently used SHM techniques include fiber optic sensors, piezoelectric or piezoresistive sensors, strain gauges, and accelerometers to monitor the mechanical deformation, vibrations, or other parameters of the structure during the operation [START_REF] Bois | Monitoring a delamination in a laminated composite beam using in-situ measurements and parametric identification[END_REF]- [START_REF] Doebling | Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review[END_REF]. However, most of these techniques can detect damage near its location therefore they must be placed near the critical zones on the structure.

To counter this, sensors network systems had also been used to triangulate the location of the damage using lamb wave propagation, but the cost, size, and weight of such a system limit their use not to mention the complex data processing required [START_REF] Andrews | Lamb Wave Propagation in Varying Isothermal Environments[END_REF]. Moreover, SHM systems attached to the surface of the composites such as optical fibers and strain gauges had a drawback of being exposed to the environmental conditions, for example, chemical, thermal, humidity, and external mechanical effect [START_REF] Schueler | Damage detection in CFRP by electrical conductivity mapping[END_REF], [START_REF] Kunadt | Messtechnische Eigenschaften von Dehnungssensoren aus Kohlenstoff-Filamentgarn in einem VerbundwerkstoffMeasuring Performance of Carbon Filament Yarn Strain Sensors Embedded in a Composite[END_REF]. That is why researchers are more focused on integrable monitoring sensors to not only monitor the overall deformation of the structure but to also monitor the internal behavior between the laminates of the composites. However, the insertion of the monitoring sensor entity in the composites is still underdeveloped and the prime focus is that it would not affect the performance of the composite structures. In previous studies, various sensors were developed and inserted inside the composites such as fiber Bragg grating, carbon nanotubes, carbon black, or carbon fibers [START_REF] Li | Health monitoring of marine composite structural joints using fibre optic sensors[END_REF]- [START_REF] Forintos | Reinforcing carbon fibers as sensors: The effect of temperature and humidity[END_REF]. Nowadays, nanomaterials had also been used to developed highly sensitive and flexible strain sensors and tested under different loading conditions for application such as multimode skin sensors, SHM of structural components and to monitor human movement etc. [START_REF] Park | Stretchable Energy-Harvesting Tactile Electronic Skin Capable of Differentiating Multiple Mechanical Stimuli Modes[END_REF]- [START_REF] Pang | A flexible and highly sensitive strain-gauge sensor using reversible interlocking of nanofibres[END_REF]. However, the use of optical sensors methods is limited because of the high cost to produce an optical fiber with fiber Bragg grating. Moreover, the use of nanomaterials as a damage sensing system is quite complex and expensive.

In comparison, carbon fibers (CF) used as a sensor because of their good electrical conductivity is a possible simple, durable, and cost-effective solution for damage monitoring in real-time [START_REF] Krucinska | Direct measurement of the axial poisson's ratio of single carbon fibres[END_REF]. CF consists of graphite-based microstructure and loading these fibers could deduce change in electrical behavior because of the change in their mechanical structure thus, depicting piezoresistive behavior [START_REF] Blazewicz | Study of piezoresistance effect in carbon fibers[END_REF]. Besides, the integration of CF in fiber-reinforced composites is quite easy because of the textile processing compatibility [START_REF] Owston | Electrical properties of single carbon fibres[END_REF]- [START_REF] Horoschenkoff | Konzeptstudie zum einsatz von elektrisch kontaktierten carbonfasern als sensor für leichtbaustrukturen aus faserverbundwerkstoff Vorträge. 14[END_REF]. The electromechanical response of Carbon fibers was the first study by Concor and Owston [START_REF] Conor | Electrical Resistance of Single Carbon Fibres[END_REF] which showed that resistance of these fibers rises linearly with the applied strain and they also studied their mechanical performance and contact resistance [START_REF] Owston | Electrical properties of single carbon fibres[END_REF]. After these studies, continuous carbon fibers had been in use as self-sensing materials in composites because of simplicity in application, high mechanical performance, and less cost [START_REF] Wen | Damage detection of carbon fiber reinforced polymer composites via electrical resistance measurement[END_REF]- [START_REF] Park | Electromechanical modeling of unidirectional CFRP composites under tensile loading condition[END_REF]. However, straightness of the filaments in the CF sensor plays a vital role to define the contact resistance and overall performance of the sensor [START_REF] Conor | Electrical Resistance of Single Carbon Fibres[END_REF], [START_REF] Todoroki | Piezoresistivity of unidirectional carbon/epoxy composites for multiaxial loading[END_REF], [START_REF] Angelidis | The electrical resistance response of continuous carbon fibre composite laminates to mechanical strain[END_REF].

In this experimental investigation, the real-time monitoring ability of the CF sensor, consisting of unidirectional carbon filaments aligned together, was examined. The sensitivity of the CF to the applied strain was calculated experimentally by the gauge factor (GF) calculation using a standalone sensor. Afterwards, the general electromechanical behavior of the CF sensor was examined up to fracture to validate its response under large strain application or during any damage that was vital to comprehend its use in high strain applications. In the next step, this CF sensor was attached to the electrodes and put in an oven to monitor the change in its electrical behavior during the change in temperature of its surroundings. The results gave interesting behavior and showed that CF sensor did not only detected the strain under mechanical loading but also showed the change in its resistance under thermal loads, which could be useful in detecting the release of thermal energy in a structure because of the presence of macro or micro cracks [START_REF] Sassi | In-situ heat dissipation monitoring in adhesively bonded composite joints under dynamic compression loading using SHPB[END_REF].

Material and Experimentation

Carbon fibers (CF) consisted of unidirectional filaments of Polyacrylonitrile (PAN) carbon fibers which were purchased from the Nanomaterials Laboratory of the University of Dayton. First, PAN Fibers are thermally stabilized at 200-300°C at room temperature and then, these fibers were carbonized in an inert environment above 1000°C. Afterwards, the surface of the fibers was etched during surface treatment. Some of the physical properties are mentioned in Table 1 for the PAN carbon fibers. SEM characterization was performed to demonstrate the filaments of CF, Figure 1.

Experimentation of standalone CF sensor under mechanical loading

CF sensor was tested under tensile load as a standalone sensor of 72 mm length and 5 mm width using the INSTRON-50 apparatus. The data acquisition system (Spider 8 manufactured by HBM) was attached through electrodes at both ends of the CF sensor to simultaneously record the variation in the electrical resistance with the applied strain, which is used to calculate its gauge factor (GF).

The sample was placed in the machine using paper support as it was difficult to place the CF sensor alone between the fixture of the machine and before the start of the test, the paper frame was cut in the middle to avoid any effect on the mechanical response of the CF sensor during the test, Figure 2.

Also, it was vital to ensure that the CF sensor was not in contact with any metallic part of the machine because it could influence its electrical response that is why all the required parts of the machine were isolated by covering with the insulation tape. It should be kept in mind that the filaments in the CF sensor were unstrained when placed between the fixture before the test and there was no slippage between the electrode and sensor connection during the test as it was accurately and properly secured between the fixtures, Figure 3. Three successful tests were conducted with the CF sensor up to fracture to comprehend its electrical behavior with the variation in mechanical performance for high strain applications. The sensor has been subjected to tensile load at a low strain rate of 2mm/min. As one will see later, the results showed repeatability in the mechanical and electrical response of the CF sensor.

Experimentation of standalone CF sensor under thermal loading

CF sensor was tested under the thermal load as a standalone sensor of 72 mm in length and 5 mm width using the CECASI oven apparatus, Figure 4. The CECASI oven system has a data acquisition system in which you can design the entire program of the thermal behavior including, temperature range, each step initial and final limit, and temperature change speed from one step to another. A paper sheet was used to place all three samples in the machine to support and isolate the CF sensor from any metallic part of the shelf. The temperature of the machine was controlled using the operating system and the data acquisition system was attached to each sensor using electrodes for starting from room temperature i.e 15°C. In each test, the temperature was changed by one degree with a rate of 0.2 °C/min, and at each degree, the temperature was kept constant for 10 mins. This step was carried out to understand the change in electrical resistance of the CF sensor in detail with defining the limit of precision.

Result discussions

Electromechanical characterization of CF sensor

The CF sensor displayed good mechanical behavior and Young's modulus and yield strength of all the examined CF sensor samples were about 94.53 MPa and 1.73 MPa on average during the standalone test, respectively, Figure 5a. Table 2 summarizes the mechanical behavior of the CF sensor, consisting of yield's strength, Young's modulus, and fracture strain. In overall mechanical behavior, each sensor sample exhibited linear elastic deformation before the start of the final fracture because of the high stiffness. CF sensor did not show any plastic deformation however, reduction in mechanical behavior was gradual due to the consecutive breakage of the filaments. Even though CF sensor showed high stiffness, but it was quite flexible because carbon filaments were held together loosely together and were combined only in both ends were electrodes were attached, Figure 5b.

Therefore, these CF sensors showed the potential to be used in strain monitoring applications without compromising their mechanical performance. Furthermore, it was observed that the damage initiation and propagation were not sudden, and the membrane was fractured gradually with the breakage of each filament. The resistance of the CF sensor was increased with the applied tensile strain which verified good correlation among its electromechanical response, Figure 6a. The sensitivity of the CF sensor was demonstrated in terms of GF by comparing the variation of resistance with the amount of applied tensile strain and calculated by using equation ( 1).

(

) 1
In this equation, △R/R o is the main constituent to define the sensitivity of the CF strain sensor as it signifies the ratio of original resistance to the variation of resistance with the applied strain ε. The GF of this sensor was calculated to be inside 10.2-10.8 range within the elastic limit, Figure 6b. In general, the GF of traditional strain sensor is around 2 while some of the multifunctional materials show higher GF values [START_REF]Strain Gauge Measurement -A Tutorial[END_REF]. For example, E. Häntzsche et al. [START_REF] Häntzsche | Characteristics of carbon fiber based strain sensors for structural-health monitoring of textile-reinforced thermoplastic composites depending on the textile technological integration process[END_REF] studied the strain monitoring behavior of sensor weaved into composites in different conditions and found the GF in the range of 0.6-1.3. However, these tests were performed on the CF sensor when embedded within a glass fiber mat composites and not as a standalone sensot. So, it was confirmed from these results the CF sensor has good strain sensitivity range and might be used for instantaneous strain monitoring of structures. Each specimen of the CF sensor presented good electrical behavior throughout the applied tensile strain, resistance changed gradually, and all samples displayed similar overall performance. The overall behavior of the CF sensor presented that, during elastic behavior the change in resistance was linear, and when the mechanical behavior of the sensor started to degrade there was a sudden increase in the resistance which reached maximum value upon fracture of the membrane, Figure 7. In addition, the sudden increase in the resistance of the sensor with the degradation of the mechanical behavior was progressing gradually to the maximum value because the carbon filaments in the sensor were breaking individual with the elongation, and with each breakage, the resistance showed variation. it was observed that when the strength of the CF sensor began to drop after achieving the peak value, its resistance started to increase linearly which confirmed good sensitivity of the sensor to detect damage initiation. All samples showed the same maximum stress and demonstrated the start of damage initiation at the almost same time. The degradation of stress in each sample showed the one by one failure of their filaments and this evolution or damage was slightly in comparison. This slight difference of failure was also observed in the evolution of respective electrical resistance of each sample and each CF sensor showed saturation of resistance to a maximum value when there were complete fracture and stress reached zero value. This confirmed its ability to use for real-time strain monitoring applications during high strain deformation of structures because the sensor showed good electrical conductance until all the filaments in it were broken.

Moreover, an increase in resistance is directly proportional to an increase in the length (elongation) of the sensor, equation ( 2)-( 3).

(

) ( 2
) 3
Where α is electrical conductivity, ρ is resistivity, L is length, A is the cross-sectional area, and R is resistance.

Electro-thermal Behavior of CF sensor

A standard two probe test was performed on three different samples between the temperature ranges of -7°C to 38°C in two sets in an oven at atmospheric pressure. Data acquisition (Spider 8 manufactured by HBM) was used to monitor the change in resistance of the CF samples and the thermocouple. A constant current of 4 mA was applied to all three samples which were randomly cut from the manufactured specimen. The applied current was kept low to prevent the self-heating of the samples and effect their electro-thermal behavior. All three samples showed an increase in resistance with an increase in temperature, Figure 8a. Moreover, each data point is provided by the error bar demonstrating the behavior of the CF sensor at each temperature change. As described earlier, the thermal load program was designed to halt the temperature change at each degree for 10 mins and error bars show the mean distribution of these data points at each degree. The overall curve profile showed the effect of temperature on the conductivity of the sensor. It showed a nonlinear parabolic increase in the resistance with the increase in temperature. All three samples showed good reproducibility in results in both cases and small error bars ensured the precision of the readings at each change of degree. Generally, it was expected to see the opposite response with the decrease in temperature but, the CF sensor showed an almost linear increase in resistance with decreasing the temperature up to 0°C, Figure 8b. From 0°C to -7°C the resistance was increasing with the decrease in temperature, but the slope of the curve was reduced. Moreover, it was observed that there was a minute decrease in the resistance of the CF sensor with a decrease in temperature and it started to increase. The filaments were freely aligned in the center while at both ends, they were combined and attached with the electrodes using insulation tape. This means that in addition to the deformation of the carbon filaments during thermal loading, the electrical contact points also played a vital role in determining the overall electrical behavior of the sensor. During the decrease in temperature, generally, the resistance should decrease which was the case in the initial points. This decrease in resistance is because of the compression of the filaments of the sensor and this could result in the decrease in the contact points between the filaments thus, reducing the multiple paths of current flow. The change in electrical contact points could increase the resistance of the sensor, Figure 9 [53]- [START_REF] Yu | Prediction of effective through-thickness thermal conductivity of woven fabric reinforced composites with embedded particles[END_REF]. This means that in addition to the deformation of the carbon filaments during thermal loading, the electrical contact points also played a vital role in determining the overall electrical behavior of the sensor. During the decrease in temperature, generally, the resistance should decrease which was the case in the initial points. This decrease in resistance is because of the compression of the filaments of the sensor and this could result in the decrease in the contact points between the filaments thus, reducing the multiple paths of current flow. The change in electrical contact points could increase the resistance of the sensor.

(5) This change is resistance can be further related to change in length or strain induced in the sample and we can use the following relation:

where GF is the gauge factor constant of the sensor, R is the original resistance of the sensor, and is the change in the resistance of the sensor with the applied strain .

By substituting equation 6 in equation 8, the change of resistance vs. temperature can give us a change in strain vs. temperature.

(

) 9
This equation can quantify the damage or strain rate induced in the specimen by the amount of energy released during the deformation or damage process. This can be sued to monitor damage in different structural materials in real-time.

Conclusions

The objective of this extensive experimental study was to develop a simple, robust, and cost-effective sensor system with high electrical conductance for multimode real-time monitoring under different loadings. This CF sensor showed viable replacement of conventional strain gauges and SHM systems because of the better flexibility, cost effectiveness, simplicity in data collection and analysis and it can be easily embedded within a sample or attached on the surface. These sensors showed high sensitivity to applied strain in the range of 10.2-10.8, were more flexible and could be easily integrated within any structure. The electro-thermal behavior showed that thermal detection with the change of resistance was because of thermal expansion and distance between the electrical connection points of straightly aligned carbon filaments in the CF sensor. Results confirmed that CF sensors in both tests reacted to the applied stimuli and showed a distinct change in their change in resistance thus, not only monitoring the deformation but also detecting the change in temperature in the surrounding at atmospheric pressure. However, further study is required to understand the precise mechanism responsible for changing the resistance of the sensors to apprehend its response under thermal loading. These CF sensors can further advance itself in real-time sensing applications within composite structures including strain monitoring, thermal degradation, and detection of failure and energy release during dynamic loading. The sensitivity of this sensor can be further tailored and

Figure 1 :

 1 SEM images of the CF sensor. (a) SEM of unidirectional filaments of Carbon aligned together (b) magnified image to show the single fiber of carbon

Figure 2 :

 2 Figure 2: Preparation of CF sensor for standalone experimental test for the GF calculation. A set of papers is used as a support and the electrode is attached on each end.

Figure 3 :

 3 Figure 3: Experimental arrangement to examine the strain sensitivity of the CF sensor.

  real-time monitoring of change in resistance with the change in temperature. A K-type thermocouple was also placed within the CECASI to verify the temperature change within the chamber. Type K thermocouple is used in this experiment. Type K is the most common type of thermocouple. It's inexpensive, accurate, reliable, and has a wide temperature range. The type K is commonly found in nuclear applications because of its relative radiation hardness. The maximum continuous temperature is around 1,100°C. It has a temperature range of -200°C to 1250°C with an error sensitivity of 0.4-0.75%. CECASI was programmed to change the thermal environment while thermocouple and the sensor were attached to the separate data acquisition system (Spider 8 manufactured by HBM) which can simultaneously record the thermal change of the thermocouple and resistance change of the CF sensor. The thermocouple was attached to one input of the acquisition system and electrodes attached to the sensor were attached to the other input of the acquisition system. Two sets of tests were performed on the CF sensors, the first test included increase in temperature up to 38°C starting from the room temperature i.e. 15°C and the second one included decrease in temperature up to -7°C

Figure 4 :

 4 Figure 4: Experimental arrangement to examine the electrical behavior of the CM sensor under thermal loading.

Figure 5 :

 5 Figure 5: Mechanical performance of the CF sensor.

  (a) Strain (ST) and Resistance (R) change (b) GF calculation

Figure 6 :

 6 Figure 6: Experimental behavior and calculation of the strain sensitivity of the CF membrane sensor. ST represents the strain of all three samples and R represent the resistance of the sample.

Figure 7 :

 7 Figure 7: Overall electromechanical response of CM sensor specimens. Stress (S) response of samples in all three tests (T) is represented by S-T while Resistance (R) of samples n all three tests (T) are represented by R-T.

Figure 8 :

 8 Figure 8: Electrical behavior of CM during thermal loading to detect thermal change.

Figure 9 :

 9 Figure 9: Demonstration of electrical contact points between the filaments of CF sensor

Figure 10 :

 10 Figure 10: Calculation of empirical relations to describe the nonlinear change in resistance with respect to temperature.

Table 1 :

 1 Physical Properties of the PAN Carbon Fibers

Density (g/cc)

1.76

Coefficient of thermal expansion (CTE)

(µm/m-°C) -0.6

Thermal Conductivity (W/m-K) 8.50

Electrical Resistance (ohm-cm) 0.00180

Table 2 :

 2 Mechanical properties of CF sensor under tensile loading

		Elastic Modulus	Fracture Strain	Yield Strength
		(MPa)	(%)	(MPa)
	Sample 1	94.212	5.16	1.72
	Sample 2	98.247	4.44	1.70
	Sample 3	91.133	4.49	1.77
	Average	94.53	4.46	1.73
	Standard deviation	3.5677	0.0354	0.0360

amplified as desired parameters by modifying the number and alignment of carbon filaments and without any significant requirements.

Conflict of interests: Authors have no conflict of interests

Authors statement: Y. Qureshi: Design and conduction of Experimentation, data curation, data analysis, investigation and data interpretation, writing-original draft preparation, M. Tarfaoui: supervision, writing-review and editing, project administration, and K. Lafdi: Samples preparation