Mrs Elisabeth Lesage

Dr Chen Shi

Dr Hongbo Lmn

Dr Shubiao Zhang

Dr Hadj Wang

Kacem Mouna

Keywords: Fatigue limit distribution, Staircase method, Vibratory strain control, Electro-dynamic shaker, High-cycle fatigue, Bootstrap sampling, Kernel Density Estimation, Bayesian maximum entropy sampling, Latin Hypercube Sampling (LHS) viii

First and foremost I am grateful to my supervisor, Asst. Prof. Leila Khalij, whose expertise was invaluable in formulating the research questions and methodology. Her insightful feedback pushed me to sharpen

Chapter 2

Experimental strain-based vibration control to obtain the fatigue limit by the staircase method

Chapter 4

A non-parametric evaluation method in staircase test for improving fatigue limit assessment

Introduction

While selecting a material for design, it is crucial to know its operating conditions. Under cyclic loading conditions, the fatigue limit or endurance limit is generally required to delimit the operating range under High Cycle Fatigue (HCF).

In this chapter, the preliminary background information on fatigue is given in order to understand the objective of the thesis work concerning the notion of fatigue limit. Section 1.1.1 introduces the fatigue properties of a material and the corresponding intrinsic variabilities leading to an estimate of the probabilistic distribution of the fatigue limit. The experimental tests being carried out in strain, Section 1.1.2 introduce the strain-stress linear relationship to provide a simple link between strain and stress in HCF. Section 1.2.1 describes the staircase methods used to estimate the fatigue limit, which is the object of this thesis. Section 1.2.2 examines the sensitivity of the staircase parameters based on the literature studies and analyses the advantages and disadvantages of the staircase method. The final section summarises the framework of the PhD study. CHAPTER 1. INTRODUCTION

Fatigue and fatigue limit

As mechanical structures move toward lightweight and complexity, fatigue design becomes more and more critical. Fatigue is the leading cause of failure for mechanical components and structures. It has been estimated that fatigue causes 90% of all service failures of metal parts [START_REF] William | Mechanical Behavior of Materials[END_REF]. It is a progressive and localised structural damage that occurs when a material is subjected to cyclic loading. Damage in metallic materials involves the initiation -commonly considered from the material's surface -and propagation of cracks by the action of the cyclic loads. The cyclic action may eventually cause the mechanical part to fail even at low load levels (below the yield stress of the material). Fatigue failure can be divided into different forms, including thermal fatigue failure, corrosion fatigue failure and vibration fatigue failure. Vibration fatigue generally refers to fatigue failure and the process of structures under a vibration environment. In recent years, many engineering structures must be serviced in a severe vibration environment [START_REF] Shi | Approach and Application of Semi-Blind Source Separation for Aero-Engine Vibration Signals Using ICA-R[END_REF], resulting in increased structural fatigue. For example, the Airbus 330 aircraft encountered engine vibration spiked up in 2011 [START_REF]Airbus A[END_REF], in which the fuel pump supply line cracked due to high vibrations. In general, vibration fatigue resistance is an essential criterion for structural design for engineering structures working in a vibrating environment.

Fatigue properties are primarily associated with the S-N curve and endurance limit [START_REF] Liu | Fatigue limit prediction of notched plates using the zero-point effective notch stress method[END_REF]. For some ferrous (iron-based) and titanium alloys, there is a threshold called endurance limit under which no fatigue failure occurs after an infinite number of repeated load cycles. For other materials (non-ferrous), this threshold does not exist, and it is therefore necessary to define a limit for a specific number of cycles. The evaluation of this fatigue limit is the purpose of the study. In practice, this parameter is hard to be determined accurately and normally it is considered as the fatigue strength which display at typically 1 × 10 6 -1 × 10 7 load cycles [START_REF] Zerbst | Applying fracture mechanics to fatigue strength determination -Some basic considerations[END_REF][START_REF] Luca | Fatigue tests for automotive design: optimization of the test protocol and improvement of the fatigue strength parameters estimation[END_REF][START_REF] Cyprian | Comparative Analysis of Fatigue Energy Characteristics of S355J2 Steel Subjected to Multi-Axis Loads[END_REF][START_REF] Ekaputra | Fatigue Strength Analysis of S34MnV Steel by Accelerated Staircase Test[END_REF][START_REF] Mei | The fatigue limit prediction of notched components -A critical review and modified stress gradient based approach[END_REF]. It must be identified by experimental fatigue tests using approaches such as staircase methods.

Fatigue properties

The fatigue properties can be defined from the S-N curve (Stress amplitude-Number of cycles to failure) as shown in Fig. 1.1. The S-N curve, also known as the Wöhler diagram, represents the fatigue property of a material, which is the most conventional design basis for fatigue analysis and is still used in engineering today. The constant amplitude fatigue test was first initialled by Wöhler on railway axles in the 1850s. In general, the classical S-N curve illustrates only a uniaxial load. Fig. 1.2 illustrates the relationship between S-N curve and fatigue limit. High Cycle Fatigue (HCF) occurs at relatively large numbers of cycles (above 1×10 4 cycles) is mainly due to elastic strain under stress amplitudes without obvious macroscopic plastic deformation (below the yield strength σ y ). Typical examples of HCF are pistons, crankshafts, connecting rods or gas turbine engines and components subject to vibratory excitation (e.g. induced by airflow on airfoils or swell on offshore structures). It is usually expressed by the S-N curve, where the amplitude of cyclic stress is plotted versus the logarithmic scale of the number of cycles the specimen can sustain before failure. The fatigue test in Very High Cycle Fatigue (VHCF) -corresponding to fatigue life region beyond 10 7 cycles -leads to a long test time, and the failure mechanism in VHCF can be quite different from that in Low Cycle Fatigue (LCF) and HCF [START_REF] Murakami | Essential structure of S-N curve: Prediction of fatigue life and fatigue limit of defective materials and nature of scatter[END_REF].

In the Fig. 1.2, S-N curve becomes horizontal at a higher number of cycles, which means that when the applied alternating stress is lower than the stress corresponding to the horizontal line, the structure can withstand an infinite number of stress cycles without fatigue fracture. The stress value corresponding to the horizontal line represents the endurance limit of the material [START_REF] Fernández-Canteli | Considerations about the existence or non-existence of the fatigue limit: implications on practical design[END_REF]. The endurance is not easy to obtain with experimental tests because it is time-consuming even if relatively high frequency is used. So in this study, a fatigue limit is sought with a staircase approach below 1 × 10 7 cycles.

From the view of engineering, the endurance limit and the fatigue limit are often confused and considered similar. However, the endurance limit is the constant stress amplitude corresponding to the asymptotic value [START_REF] Mei | The fatigue limit prediction of notched components -A critical review and modified stress gradient based approach[END_REF]. In some cases, the fatigue limit is chosen at a specific large number of cycles (e.g., 1 × 10 6 cycles) to reduce the testing time or to depend on the structural design. Both parameters are important material properties for various engineering predictions on fatigue [START_REF] Schijve | Fatigue of structures and materials in the 20th century and the state of the art[END_REF], in particular, to apply the mean stress correction through Goodman's method, for example [START_REF] Susmel | Fundamentals of fatigue assessment[END_REF]. In this dissertation the fatigue limit is defined: fatigue strength (stress amplitude) for a given number of cycles targeted 1 million from the S-N curve [START_REF] Liu | Fatigue limit prediction of notched plates using the zero-point effective notch stress method[END_REF][START_REF] Mei | The fatigue limit prediction of notched components -A critical review and modified stress gradient based approach[END_REF][START_REF] Nadot | Fatigue from Defect: Influence of Size, Type, Position, Morphology and Loading[END_REF][START_REF] Endo | A practical method for fatigue limit prediction in ductile cast irons[END_REF]. In body-centred cubic materials (bcc, such as steel and titanium), a pronounced fatigue resistance can often be determined (endurance limit). In contrary, face-centred cubic materials (fcc, such as aluminium and copper) do not have a specific limit and will eventually fail even from small stress amplitudes, the S-N curve drops over the entire load cycle range for these materials [START_REF] Boyer | Atlas of Fatigue Curves[END_REF]. Therefore, engineers try to design their parts to keep them under the fatigue limit during work.

However, the fatigue limit assessment has never been an easy task, as fatigue has considerable dispersion due to variability in the fracture process [START_REF] Schijve | Fatigue predictions and scatter[END_REF][START_REF] Ying | A variance reduction technique for long-term fatigue analysis of offshore structures using Monte Carlo simulation[END_REF] and uncontrolled test conditions in a rigorous manner, which is independent of the experimenter.

Pascual et al. [START_REF] Francis | Estimating Fatigue Curves with the Random Fatigue-Limit Model[END_REF] established a random fatigue limit model to describe the variability increment in fatigue life around the fatigue limit. As a result of inherent variation in fatigue data [START_REF] Svensson | Random Features of the Fatigue Limit[END_REF], the S-N curve and the fatigue limit must be drawn on the distribution. Most approaches evaluate the fatigue limit in the form of a distribution (Cumulative Density Function (CDF) or Probability Density Function (PDF)) to estimate a mean or median value of the fatigue limit but also a variance or standard deviation to give a measure of its variability. The distribution is also used to estimate the probability of failure or success of an experiment repeated several times. The median S-N curve and its dispersion (dotted curves) are schematically represented in Fig. 1.3. During real fatigue tests, the probabilistic fatigue limit performed at the given number of cycles (N L ) [START_REF] Baptista | Probabilistic S-N curves for constant and variable amplitude[END_REF], which is restricted to the HCF region. In numerical design, the fatigue limits obtained from uniaxial tests (in torsion, bending or tension) are essential in evaluating fatigue strength when the structure is subjected to multiaxial loads. The stress-based criteria is expressed as inequality between the equivalent shear stress and a function of the fatigue limits. When the structure is submitted to random loads, the function of the fatigue limits, which is expressed as a probability distribution, can be used in a reliability study. Thus, Lambert et al. [START_REF] Lambert | A probabilistic model for the fatigue reliability of structures under random loadings with phase shift effects[END_REF] proposed a probabilistic model from the Sines criterion using the extreme value theory to determine the equivalent shear stress distribution. The fatigue limit function can be obtained from experimental tests to complete the evaluation by the constraints-resistance approach. This work is in line with this objective because it is not easy to reach these distributions owing to the requirements of testing time and a large number of specimens.

The methods of obtaining the fatigue limit can be classified as follows:

(1) Probabilistic modelling of S-N curves. The statistical fatigue limits are derived from the probabilistic curves. However, this method is inaccurate because the data are distributed in the whole S-N curve rather than around the fatigue limit. (2) Numerical methods. Firstly, Ray-projection and Parallel-projection methods [START_REF] Lin | Evaluation of the staircase and the accelerated test methods for fatigue limit distributions[END_REF].

These two methods extrapolate the HCF property from LCF test data to avoid long-time tests. Lin [START_REF] Lin | Evaluation of the staircase and the accelerated test methods for fatigue limit distributions[END_REF] proved the parallel-projection method provided reasonable results for fatigue limit distribution. However, this method gradually lost its advantage with the development of HCF test equipment. Secondly, Monte-Carlo simulation (MCS) method [START_REF] Zhao | Improved measurement on probabilistic fatigue limits/strengths by test data from staircase test method[END_REF] aims to generate several fatigue limit values with different probabilities and confidence values from one reference fatigue limit according to a specific distribution. Thirdly, many researches [START_REF] Liu | Fatigue limit prediction of notched plates using the zero-point effective notch stress method[END_REF][START_REF] Mei | The fatigue limit prediction of notched components -A critical review and modified stress gradient based approach[END_REF][START_REF] Nadot | Fatigue from Defect: Influence of Size, Type, Position, Morphology and Loading[END_REF][START_REF] Cetin | The fatigue limit: An analytical solution to a Monte Carlo problem[END_REF] have proposed a fatigue limit prediction model considering the notch or defect in the structure. The numerical methods are based on previous fatigue information rather than a direct measurement. (3) Experimental testing. In fact, the fatigue limits deduced from random fatigue performance should be verified by true test data. The experimental fatigue test is the main tool for assessing the fatigue limit of a material [START_REF] Li | Evaluation of the Methods for Estimating the Fully Reversed Unnotched Fatigue Limits of Steels[END_REF]. Several experimental evaluation test strategies have been proposed in the literature, mainly including staircase method [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF], step stress method [START_REF] Nicholas | Step loading for very high cycle fatigue: STEP LOADING FOR HCF[END_REF], locati method [START_REF] Lipski | Determination of Fatigue Limit by Locati Methodusing S-N Curve Determined by Means of Thermographic Method[END_REF], prot method [START_REF] Boresi | An appraisal of the prot method of fatgiue testing Part I[END_REF], probit method [START_REF] Kuroda | Historical Review on Origin and Application to Metal Fatigue of Probit and Staircase Methods and Their Future Prospects[END_REF]. These methods are compared in Tab. 1.1. In addition, an experimental thermal method [START_REF] Ricotta | Comparison of Experimental Thermal Methods for the Fatigue Limit Evaluation of a Stainless Steel[END_REF] can be used to measure the temperature change during the fatigue process to reach the limit. However, this method requires infrared thermography. Despite its drawbacks (mentioned in Section 1.2.2), many studies try to improve the estimation of the probabilistic fatigue limit by staircase methods. This method has advantages in both experiments (less number of specimens and less test time) and statistics (more efficient). This work, therefore, is focused on the staircase test method presented in Section 1.2.

Strain-stress relationship in HCF

High Cycle Fatigue (HCF) is a type of fatigue caused by small elastic strains under a high number of load cycles before failure occurs. The stress comes from a combination of mean and alternating stresses. Let us consider the case of a proportional load with constant amplitude calculated from the minimum and maximum stress values of A power formula relating stress amplitude with the number of cycles, proposed by Basquin in the 1910s [START_REF] Basquin | The exponential law of endurance tests[END_REF], is expressed as:

σ a = C 1 × N C 2 f (1.2)
where, C 1 and C 2 are constant related with material; σ a is the stress amplitude and N f is number of cycles to failure.

The S-N curve and the models for describing HCF are widely used to the present day. Nevertheless, strain-based formulations are required in this study to consider the measurement means used during the tests (strain gauges, for example). When a deformable solid (material) is subjected to a uniaxial loading state with a fully reversed cyclic stress (or strain) loading in the linearly elastic strain range, the stress and strain correlate linearly with Young's modulus E. For example, the Fig. 1.4 illustrates cyclic strain ε(t) as a function of cyclic stress σ(t). It is evident that the strain cycle (0-1-2-... -11-12) is the same as the stress cycle (0-1-2-... -11-12), and the stress-strain curve can be simplified as a straight line in the elastic domain. The fatigue limit lies in the high cycle fatigue domain, so the linear relationship between strain and stress is also valid. Thus, the specimens subjected to a load close to the fatigue limit have an elastic behaviour. Therefore it can be assumed that the specimens follow the general Hooke's law:

σ ij = C ijkl ε kl (1.3)
where σ ij and ε kl are the engineering stress and strain, respectively. C ijkl is the elasticity tensor that depends only on the material. The fatigue Stress Concentration Factor (SCF) can be included in the analysis by multiplying the stress calculated in Eq. 1.3 by K f given in Appendix A.2.3. Fig. 1.5 shows a typical log -log plot of strain amplitude versus the number of cycles to failure. The total strain (ε) can be considered as the sum of elastic strain (ε e ) and plastic strain (ε p ). The total strain occupies the main part in LCF while the plastic strain occupies the main part in HCF. Therefore, the strain-number of cycles to failure (ε -N ) curve can be divided into two parts according to elastic strain and plastic strains. The plastic strain can be expressed by the Manson-Coffin law:

ε p = ε f × N c f (1.4)
where ε f and c are fatigue ductility coefficient and the constant, respectively. The Basquin's law (given in stress in Eq. 1.2) also can be used to express the elastic strain as follows:

ε e = σ f E × N b f (1.5) 1.2. STAIRCASE TEST
where σ f and b are the fatigue strength coefficient and a constant obtained from the slope of the log-log S-N curve, respectively. Since the stress in HCF tests is usually within the elastic range, the present work focuses on the fatigue limit by considering only elastic strain under small deformation. If we focus on the range within fatigue limit, ε p = 0 and ε = ε e . Thus, the strain and stress have a linear relationship with Young's Modulus E. It can be concluded that stress control and strain control around the fatigue limit range can reach the same results when the same test conditions are considered. This strong assumption is important for this work because it provides us to use strain control in Chapter 2 to evaluate the fatigue limit.

Staircase test

The fatigue properties of materials can be appraised on statistical grounds. The fatigue limit used in the design must be fully characterized by its statistical nature. The probability fatigue limit can be deduced from probabilistic S-N curve. Tests are specially conducted to characterize fatigue limit distribution.

The specimens are tested at various stress levels, and the corresponding number of cycles (fatigue life) is evaluated. The experimental results are usually prepared as Fig. 1.6. As the Fig. 1.6 shows, it is natural that the fatigue limit is deduced from S-N curve. The S-N curve can obtain a fatigue limit value but cannot provide a statistical distribution. Considering that the fatigue limit can be seen as a random value, the staircase method [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] is much more suitable for the fatigue limit estimation accuracy.

Staircase test protocol

Dixon and Mood [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] proposed the classical method for sensitivity tests in 1948, which is the staircase method. This method is widely used in machinery, medical, pyrotechnics, etc. Both standard ISO12107 [START_REF]Metallic materials-Fatigue testing -Statistical planning and analysis of data[END_REF] and the standard MIL-STD-331D [START_REF]D. Fuzes, ignition safety devices and other related components, environmental and performance tests for[END_REF] recommend this test procedure for experimental test.

As the staircase procedure, fatigue tests are carried out sequentially on specimens subjected to a constant amplitude stress cycle until a predetermined number of cycles N L . The first specimen is tested at an initial load amplitude selected arbitrarily. Suppose the specimen "survival" (runout) until N L cycles. The load amplitude applied to the next specimen is increased by a step size d. Conversely, the specimen is marked as "failure", and the load amplitude for the next specimen will be decreased by a step size d. The step size is usually constant during the entire experiment process. This procedure is repeated in sequence, with load levels being increased and decreased in increments until the number of specimens is reached. Therefore, the staircase approach provides a reasonable estimate of the median fatigue limit because about half of the specimens fail and the others do not [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF].

Let n be the sample size of a staircase experiment, and the specimens that to be tested are denoted as random variables X = [x 1 , x 2 , . . . , x i , . . . , x n ]. The X are chosen from a discrete and finite set of stress amplitude S a , S a = [S a,1 , S a,2 , . . . , S a,j , . . . , S a,jmax ]. The S a is discrete by step size d. After a specimen is tested in x i = S a,j , the next specimen must be treated one load level up or down as the current specimen. The responses of each specimen are denoted as the "0" survival and "1" failure. Repeating the same staircase rules turns a finite set of stress levels into random variables.

The staircase test methodology was extensively used to evaluate fatigue limit distribution due to its three outstanding features. Firstly, by its very nature, the staircase test tends to concentrate data near the median, leading to a high accuracy estimation of the median. Secondly, the staircase sequence design can result in "30 %to 40%" savings in the number of specimens compared with creating a whole S-N curve. Thirdly, it is the relative simplicity of the statistical analysis of staircase data.

However, the staircase method is not always working well because the standard deviation estimation is greatly affected by step size in the staircase method, which must be constant during the test, and leads to low standard deviation estimation accuracy. With this drawback, some simulation studies have shown that deviation bias is a function of both step size (d) and sample size (n) in staircase testing [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF]. The staircase method cannot eliminate or avoid this problem. However, it can minimize the error of the standard deviation estimation value in optimizing the experimental design [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF], improving the data processing [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] and creating the correct coefficient [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF]. The detailed background are presented in Section 4.1 and Section 5.1.

STAIRCASE TEST

Staircase parameters

Four parameters are input for the staircase method, including initial stress, stress step, number of specimens and the predetermined number of cycles.

(1) Initial stress (S a,0 ) Generally, the engineer could make a qualified guess about the expected fatigue limit based on the experience with the mechanical properties of the material that are derivatives from the literature and previous tests. Despite the fact that the engineer's experience with a particular component and the results from estimations may differ, it is indispensable to select the initial level. Some research [START_REF] Rabb | Staircase testing -confidence and reliability[END_REF] have studied the effect of initial stress on the staircase outcomes. Usually, the starting stress level should preferably be taken as close to the mean threshold stress as possible [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF]. The initial level, which is a highly realistic boundary condition, is integrated into the simulation model. Roué et al. [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] choose a "median-3/2d" as an initial value in the simulation study to reduce the number of specimens to approach the median fatigue limit. Müller et al. [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] firstly proposed a function for choosing an initial level underlying the Lognormal distribution.

(2) Step size (d)

Step size is a significant parameter in the staircase method. The first reason is that if the value of the step is too small, there will be continuous survivals or failures, resulting in wasted specimen and time. Conversely, if the interval value is too large, the estimated standard deviation will be much more significant. In the conventional staircase method, the increment must be constant in one test. The other more crucial reason is that increments greatly influence standard deviation estimation.

In many researches, the increment was selected as a proper integer value without deliberate research such as 5 MPa, 10 MPa or 20 MPa.

A number of studies investigated the parameters in the staircase method. In the assumptions of Dixon-Mood, the increment should keep constant in every stress level, and the increment should be in the range of 0.5s-2s [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF], where s donate the standard deviation of fatigue limit. Yoshimoto [START_REF] Yoshimoto | Fatigue test by staircase method with small samples[END_REF] recommended that the step size should be as close to the standard deviation of fatigue limit as possible in a small sample size test and applied linear regression analysis of the experimental S -logN curve to find a "standard deviation". "5% criterion" [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF] is a consensus and an engineering experience. When the fatigue strength determined by the conventional fatigue test method is known, the fatigue strength within 5% can be taken as the increment. However, this assumption relies on the constant increment being less than twice the actual standard deviation of the tested population. Grove [START_REF] Grove | A comparison of two methods of analysing staircase fatigue test data[END_REF] assumed that it is more robust to the choice of the increment in the range s -2s, where s is the standard deviation of the log-stresses. Pollak et al. [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] for a range of increment (Fig. 1.7) and found that using larger step sizes in the 1.6s -1.75s range can significantly reduce estimation bias. Çalişkan et al. [START_REF] Çalişkan | Determining the endurance limit of AISI 4340 steels in terms of different statistical approaches[END_REF] advised to use stress increment around 0.5s-1.5s.

Figure 1.7: Effect of the increment on the standard deviation estimation [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] So, the standard deviation is required prior to the test to determine the increment. Standard deviation is available from another testing of the material or similar material to provide a rough initial estimate of standard deviation.

(3) Number of specimens (n) In the early research, the sample size of the staircase test should be large enough (40-50 specimens) for the staircase method. In recent years, staircase tests normally consist of 10-30 specimens. Pollak et al. [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] found that the method is robust for sample numbers as small as 5-10 specimens. We believe that too many samples will consume much test time and money, while too few samples do not yield accurate results and affect the subsequent probabilistic analysis. This observation is supported by the study of Roué et al. [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF], which proved that a small sample size would greatly reduce the confidence of fatigue limit. Logically, the uncertainties decrease when the number of specimens increases for the same distribution (same fatigue test). ISO 12107 [START_REF]Metallic materials-Fatigue testing -Statistical planning and analysis of data[END_REF] and Strzelecki's research [START_REF] Strzelecki | Experimental method for plotting S-N curve with a small number of specimens[END_REF] demand that at least 15 specimens to estimate the mean and the standard deviation for exploratory research. Generally, the sample size is small, for example, less than 20.

The nominal sample size is used in both simulation and experimental studies to reduce the effect of starting load level. The nominal sample size [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF][START_REF] Zhao | Probabilistic measurements of the fatigue limit data from a small sampling up-and-down test method[END_REF] is the number of specimens after and including the first pair of tests with opposite results. The number of specimens in this study (n) donates the nominal sample size.

(4) Number of cycles (N L )

In practice experiment, it is hard to reach an infinite number of load cycles, and the 1.2. STAIRCASE TEST fatigue limit normally is considered as the fatigue strength which displays at typically 1 × 10 6 or 1 × 10 7 load cycles [START_REF] Zerbst | Applying fracture mechanics to fatigue strength determination -Some basic considerations[END_REF][START_REF] Luca | Fatigue tests for automotive design: optimization of the test protocol and improvement of the fatigue strength parameters estimation[END_REF][START_REF] Cyprian | Comparative Analysis of Fatigue Energy Characteristics of S355J2 Steel Subjected to Multi-Axis Loads[END_REF][START_REF] Ekaputra | Fatigue Strength Analysis of S34MnV Steel by Accelerated Staircase Test[END_REF]. In this study, we have chosen 1 × 10 6 cycles for defining the fatigue limit.

In the literature, fatigue limit tests based on the staircase method in recent years are summarized in Tab. 1.2.

In Tab. 1.2, m is the median (equal to mean in Normal distribution), s is the standard deviation, N L is fatigue life for defining the staircase test method, and n is the number of specimens. The number of specimens is less than 30 in most studies. All tests were conducted considering stress levels, and results were calculated by the classical Dixon-Mood (DM) method (which is described in [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF]) for Normal distribution unless otherwise specified.

In summary, the staircase methods have four outstanding features: (1) The simple protocol is the most obvious superiority. Each specimen was tested at given stress for a specified number of cycles or until failure, and only adjusted stress load for the different specimens. [START_REF] Shi | Approach and Application of Semi-Blind Source Separation for Aero-Engine Vibration Signals Using ICA-R[END_REF] This method tends to concentrate data near the median by its nature [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF]. (3) In the following, the staircase method requires fewer tests. Compared with the Probit methods (See Tab. 1.1), it saves the tested number of specimens 30%-40% [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF]. (4) Thanks to Dixon and Mood's research [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF], there is the relative simplicity of the basic statistical analysis of staircase data.

However, the staircase method has three defects: [START_REF] William | Mechanical Behavior of Materials[END_REF] The staircase method test does not have mathematical optimization characteristics, and the accuracy of the estimation under the same sample size remains to be further increased, or the sample size needs to be further reduced under the same estimation accuracy. [START_REF] Shi | Approach and Application of Semi-Blind Source Separation for Aero-Engine Vibration Signals Using ICA-R[END_REF] The stress step size d should be estimated before the staircase test. If the step size is too large, the probability of generating invalid test data increases, increasing the test uncertainty.

(3) The standard deviation estimation is systematically less accurate and lower than the true value. Therefore, this study carried out the experimental fatigue test by staircase method and improved it in terms of post-processing and test strategy. Normal distribution: m: median(mean) s: standard deviation 1.3. SUMMARY

Summary

Fatigue characterization has been a research topic in mechanical structure design for decades. A brief review of the fatigue limit and staircase method is presented in this chapter. Section 1.1 introduces fatigue limit -with its variability property -and describes the linear elasticity in the HCF test, which is the base of the strain control method proposed in this work. In Section 1.2, the staircase protocol and parameters sensitivity analysis are summarized. The studies below are carried out based on the advantages of the staircase method and aim to reduce the disadvantages. Chapter 2 In this chapter, an experimental strain-based vibration control test method is introduced and conducted with an electro-dynamic shaker . This chapter details the use of a staircase test procedure on a vibration bending bench, including parameter selection and a fatigue limit estimation with the classical Dixon-Mood approach. The test results with the staircase method obtained in this chapter offer a dataset for uncertainty quantification and statistical study.

Experimental test

Chapter 3

The post-processing leads to one distribution after one staircase test. We want to evaluate the uncertainty from multi-staircase tests on the same experimental test system. Bootstrap is used to characterize the mean and standard deviation of the fatigue limit distribution from staircase data, as well as the uncertainty associated with the staircase test method. Using the bootstrap method, we also compared the fatigue limit distribution of the DC01 from different kinds of specimens.

Chapter 4 The use of different methods from the literature led us to investigate how to approach the statistical treatment to improve the standard deviation estimation of the fatigue limit distribution. We reached a new method based on Kernel Density Estimation, that we have detailed in this chapter. A simulationbased investigation is performed using several probability distributions with different coefficients of variation.

Chapter 5 In the logic of making improvements to the staircase methods, we chose to focus on the use of a test design process based on the Bayesian theorem. In an attempt to improve the results, we also proposed to include Latin Hypercube Sampling (LHS) regarding both the scientific consideration and the engineering requirement. The advantage of using the Bayesian theorem in the staircase method is that it eliminates the step size and makes the most of the previous test information to decide on the following test. LHS is also involved in preventing similar results from the same Bayesian Maximum Entropy Sampling (BMES).

The results provide that the optimal staircase method has a faster convergence than the conventional staircase method.

This dissertation finally ends with general conclusion and perspective work.

Chapter 2

Experimental strain-based vibration control to obtain the fatigue limit by the staircase method

This chapter highlights the possibility of strain control with a vibration fatigue bench. The combination of a constant strain level and staircase methods provides a quick evaluation of the fatigue limit distribution.

Low carbon steel specimens with a reduced section were selected to carry out experimental tests. A staircase test procedure with a vibration bending bench is detailed in this work, including parameter selection. The results highlight the efficiency of the strain-controlled staircase method in reaching the fatigue limit.

Section 2.1 is a non-exhaustive overview of the fatigue test machines. Section 2.2 describes the specimen, the experimental vibratory bench and the instruments. Section 2.3 presents a methodology for strain control and a comparison with traditional acceleration control. Section 2.4 gives the validation for the experimental data and introduces the next chapter, which aims to provide improvements for the estimation of the statistical parameters of the fatigue limit distribution. This part has been published in an international journal [START_REF] Shi | Experimental strain-based vibration control to obtain the fatigue strain limit by the staircase method[END_REF].

Fatigue test machines and their control

As pointed out by Shawki [START_REF] Galal | A Review of Fatigue Testing Machines[END_REF], fatigue testing machines can be classified according to different criteria depending on the requirements. The method of load application is an essential criterion because it determines the amplitude of the applied force and the test duration. Usually, hydraulic machines are chosen for their ability to apply relatively high force amplitudes compared to machines using high frequencies to excite the components (such as electro-dynamic, electromagnetic, or ultrasonic systems). Other criteria are essential such as the type of loading (tension, bending, the combination of bending and torsion, ...) or the need to control by strain or stress.

There are three representative clusters of testing machines usually adopted to perform fatigue tests of structures: servo-hydraulic machine, electro-dynamic shaker, and ultrasonic testing system. The difference between these three machines is listed in Tab. 2.1. From Tab. 2.1, the main difference between these three kinds of testing systems is the working frequency range. In one way, the ultrasonic fatigue tester presents an excitation frequency of more than 20kHz and is mainly used for VHCF test. In another way, the load-controlled servo-hydraulic machine and electro-dynamic shaker are used in HCF tests. However, hydraulic machine may leads to a long time to achieve a large number of cycles around the fatigue limit due to the relatively low load frequency . Whereas, electro-dynamic shaker has a significant advantage in saving the fatigue test time since they work on high load frequency. For example, it costs about 93 hours for 30 Hz in the servo-hydraulic machine to reach 1×10 7 cycles, while only 13 hours for 220 Hz in electro-dynamic shaker. Since testing such large numbers of cycles is time-consuming, it is desirable to accelerate the tests by electro-dynamic shaker to 2.1. FATIGUE TEST MACHINES AND THEIR CONTROL reach a high number of cycles.

These systems offer different ways of applying the load. Hydraulic machine can directly perform fatigue testing in force (to control stress) or displacement (to control strain) on the structure. Electro-dynamic shaker is performed chiefly with baseexcitation to obtain a large deflection of the structure with small input excitation. Furthermore, ultrasonic testing system control the displacement of the specimen through a piezoelectric actuator. The load application determines that the response of the servo-hydraulic machine and ultrasonic fatigue tester are well-controlled, but not for electro-dynamic shaker.

To sum up, the servo-hydraulic machine has reasonable constant load control but leads to a lengthy test duration. The electro-dynamic shaker has an advantage in reducing test time but has drawbacks in indirect control. The ultrasonic fatigue tester works in VHCF and must consider other problems like frequency effect and thermal effect in application [START_REF] Peng | A Brief Review of the Application and Problems in Ultrasonic Fatigue Testing[END_REF].

The difficulty of using electro-dynamic shaker in the fatigue test lies in control. The shaker introduces the motion of the basement by controlling acceleration, velocity or displacement through a closed-loop system. However, the major drawback of baseexcitation is that the stress or strain response is widely dispersed. Khalij et al. [START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF] used the base-excitation in acceleration to establish a fatigue diagram corresponding to the strain amplitude versus the number of cycles, which highlights the difficulty of controlling the strain amplitude levels for the same acceleration levels.

Several other works [START_REF] Machado | Spectral formulated modelling of an electrodynamic shaker[END_REF][START_REF] Julian | An Overview of Fatigue Testing Systems for Metals under Uniaxial and Multiaxial Random Loadings[END_REF] have proposed the development regarding the objection to keeping stress/strain steady for base-excitation of electro-dynamic shaker test systems. Among these, a simple approach is to use force as the input excitation. Hooreweder et al. [START_REF] Van Hooreweder | Fatigue strength analysis of notched aluminium specimens using the highly stressed volume method[END_REF] used an electro-shaker which directly controls the force through a stiff stinger at the free end of the specimen instead of base-excitation. However, force control requires a preload for the sensor and loses the advantage of resonant excitation with the shaker. Another typical approach is to use output sensors to calibrate the local stress (strain) in the fatigue zone. By measuring displacement, Xue et al. [START_REF] Xue | Fatigue life assessment of a high strength steel 300 M in the gigacycle regime[END_REF] focused on fatigue limit assessment and used an optical displacement sensor to measure displacement at the specimen free edge and then measure the applied stress. Morrissey et al. [START_REF] Morrissey | Staircase testing of a titanium alloy in the gigacycle regime[END_REF] performed the fatigue test by calibrating the displacement to give a feedback loop during testing to control the strain actively. Gautrelet et al. [START_REF] Gautrelet | Linearity investigation from a vibratory fatigue bench[END_REF] used the linear relationship between the response in strain or velocity and the excitation in acceleration to calibrate the strain level of the fatigue zone. Kim [START_REF] Kim | Experimental spectral damage prediction of a linear elastic system using acceleration response[END_REF] proposed a spectral damage prediction method based on acceleration response without any strain data, which is based on the spectral relationship between the energy of acceleration and fatigue damage. George et al. [START_REF] George | Development of a novel vibration-based fatigue testing methodology[END_REF] used a laser sensor to measure the velocity of the specimen to calibrate to the maximum strain. Similarly, Ellyson et al. [START_REF] Ellyson | Characterization of bending vibration fatigue of WBD fabricated Ti-6Al-4V ». en[END_REF][START_REF] Ellyson | Characterization of bending vibration fatigue of SLM fabricated Ti-6Al-4V ». en[END_REF] CHAPTER 2. EXPERIMENTAL STRAIN-BASED VIBRATION CONTROL TO OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD used a closed-loop system to maintain a steady force and frequency of the base excitation, which is controlled against the displacement of a point on the sample. Česnik et al. [START_REF] Česnik | Uninterrupted and accelerated vibrational fatigue testing with simultaneous monitoring of the natural frequency and damping[END_REF] designed a closed-loop control system from two accelerometers as outputinput based on a Y-shaped specimen to actually controll the stress in the fatigue zone, but no real-time was used. However, calibration to strain may be impossible if the sensor voltage exceeds the test software's limit in practice. Moreover, the relationship of output (displacement/velocity/acceleration) -strain/stress is susceptible not only to structure, such as dynamics characteristics, but also to other factors such as the material and the environmental temperature [START_REF] Kim | Experimental spectral damage prediction of a linear elastic system using acceleration response[END_REF]. In brief, conventional experimental calibration measures strain rather than control of strain.

The literature dealing with the vibration-induced fatigue control method still focuses on indirectly controlling for stress/strain. Since the strain assessment depends on the frequency response magnitude, strain control is essential to reach the fatigue parameters.

In previous works, most fatigue tests were controlled by strain using the hydraulic machine. For example, the strain control fatigue test normally aims at stress-strain [START_REF] Presas | Fatigue life estimation of Francis turbines based on experimental strain measurements: Review of the actual data and future trends[END_REF] and LCF behavior. A systematic study in this area is the work of strain-life curves. Procházka et al. [START_REF] Procházka | Strain controlled cyclic tests on miniaturized specimens[END_REF] dealt with strain-controlled cyclic tests, the results of which enabled the construction of Manson-Coffin curves and assessment fatigue parameters. Williams [START_REF] Williams | A practical method for statistical analysis of strain-life fatigue data[END_REF] studied the development of accurate statistical strain-life curves, to predict LCF and HCF based on plastic strain and elastic strain. Carrion et al. [START_REF] Carrion | Strain-based fatigue data for Ti-6Al-4V ELI under fully-reversed and mean strain loads[END_REF] published the experimental data about the strain-controlled fatigue tests on TA6V at different strain ratios and various strain amplitudes.

In this study, we want to control strain in HCF vibration fatigue test. This section presents the difficulties encountered related to the use of a vibration system and the excitation at the resonant frequency of the specimen.

Experimental setup

Specimen design and material

For experiments, specimens were manufactured from plates using a wire electrical discharge machine (wire-cut EDM). A stress raiser zone was designed in the middle of the specimen to localize the crack far from the clamp. The quality level of the finished machining surface is better on the reduced section. More details about the specimen dimensions are shown in Fig. From the FEM simulation, the resonant frequency of the first bending mode is 211.35 Hz. To avoid the effect of the second mode, the specimen has been designed in order to obtain the second mode away from the first. Besides, the Frequency Response Function (FRF) are obtained by harmonic analysis range of 100 Hz-2000 Hz under an acceleration of 3 g. The FRF of the strain in the central point (the location of strain gauge) is shown in Fig. 2.4. In this figure, only two peaks represent the first and third 2.2. EXPERIMENTAL SETUP modes. The plan mode and the torsion mode are not visible because of the normal measurement. Hence, the first bending leads to the largest deflection and, therefore, more damage in the reduction area. Therefore, the first mode is chosen for the fatigue tests.

Test bench

The electro-dynamic shaker has the capability of the fatigue test with a high number of cycles, such as 1 × 10 6 cycles, in a short time. The Laboratory of Mechanics of Normandy (LMN) has acquired a vibration-based bending fatigue bench which consists of a motion table driven by a shaker presented in Fig. 2.5. The air-cooled electro-dynamic shaker is connected to the test specimen through a shaker table. A closed-loop vibration control ensures real-time adaptive control of tests. The experiments are performed on low carbon steel specimens subjected to cyclic sine excitation in bending deformation. Pictures of the machine are shown in Fig. 2 For the Dongling GT500M shaker with a horizontal mobile table, the acceptable working conditions are listed in Tab. 2.4. In this test, the instrumentation and software used to capture the signals are:

(1) A PCB 352A24 accelerometer with sensitivity 100 mV/g is fixed on the head of table .  (2) A HBM 350GE LM11 micro-measurement strain gauge with the ability of high resistance to alternating loads is glued on the stress concentration area of the specimen. The sensitivity is 0.5 mV/(µm/m). The performance of the strain gauge during the fatigue tests is discussed in Section 2.3.2 because the strain control has a great significance in this study. The shape and dimension of the strain gauge used in this study are presented in Fig. 2.7. The roles of the strain gauge and the accelerometer have been permuted during the study to have : -The accelerometer serves as the table control sensor to excite the specimen. In this case, the gauge measures the strain response of the specimen.

-The gauge is a control sensor of the specimen strain. In this case, the accelerometer becomes a device for measuring the acceleration that compensates for the displacement of the table according to the imposed strain. In both cases, the two sensors can have a control role but can also be used to measure. Indeed, there is always a difference between the control set point and To carry out the fatigue test on the vibration table, the excitation frequency is selected near to the first resonance frequency of the specimen, and the specimen is forced to produce an enormous deflection response due to the resonance amplification.

Variability in the experimental test

Before fatigue testing, it is necessary to identify the factors that affect the test results. Since the main objective is to reduce uncertainty, this part helps to categorize these factors and make necessary assumptions. Most importantly, as described in Section 1.1.2, the strain and stress are linear in this high-cycle vibration fatigue testing.

To prevent uncertainty of the test system, the following arrangements have been adopted: (1) All tests were performed at constant room temperature; (2) The connection of the test specimen was ensured by clamp on accurate surfaces [START_REF] Appert | Development of a test bench for vibratory fatigue experiments of a cantilever beam with an electrodynamic shaker[END_REF] with a constant tightened torque at 25 N • m.

Alignment of testing machine and specimens

The clamping in the vibration bench is not precise as the hydraulic machine. Even if the clamp is tightened with a torque wrench, the positioning of the specimens can have an impact on the resonance and damping. Moreover, the choice of the first resonance mode for testing is problematic, as shown in the work of Appert et al. [START_REF] Appert | Development of a test bench for vibratory fatigue experiments of a cantilever beam with an electrodynamic shaker[END_REF]. That is why a reduced section has been added away from the clamp to reduce its effect.

Specimens mass

The test results are highly dependent on the specimens. However, the specimens have variability in dimensions even if they come from the same manufacturing batch. Therefore, these dimension variations lead to mass (or density) variations, and the effect is not negligible when dynamic behaviour is considered.

Material and micro-structure The specimens being extracted from a plate, variability of the micro-structure and defects contained in the material, including size, type, distribution and morphology, have an effect on the fatigue behaviour [START_REF] Nadot | Fatigue from Defect: Influence of Size, Type, Position, Morphology and Loading[END_REF][START_REF] Ahmadi | Experimental high cycle fatigue testing and shape optimization of turbine blades[END_REF].

Crack initiation For the damage in HCF, there are micro-damages on the surface of the deformed solid (Material) that lead to micro-cracks through slippage (crack initiation). Then the micro-cracks develop perpendicular to the direction of the load (crack propagation). It is assumed that cracks are always initiated from the side face of the reduced area and grow in the transverse direction.

Linearity of the test system in acceleration control

To prove that no other parameter affects the system (for example, plasticity), sine sweep tests at different acceleration levels were performed to validate the linearity of the test system. The sine sweep frequency is around the first resonant frequency, which is 190 Hz-220 Hz with a sweep speed of 0.3 Hz/s. From CES EduPack software [START_REF]Ansys GRANTA EduPack software[END_REF], we know that the theoretical fatigue limit of the studied steel does not exceed 1800 µm/m. Therefore, the interest range can be focused on acceleration levels from 1 g to 4 g every 0.5 g. The FRF, representing the ratio in frequency between the response and the excitation, are plotted in Fig. 2.8. These figures reveal a shift of the resonant frequency from 190 Hz to 220 Hz as well as a variation of the system damping factor in a range from 2.494 × 10 -3 to 5.387 × 10 -3 increasing with the acceleration levels. Besides, the maximum strain in Fig. 2.8 is 1684 µm/m, which corresponds to a stress close to 340 MPa. This stress is lower than the yield strength (Tab. 2.2) of steel DC01, which confirms that the material works in the elastic region in this acceleration range.

Gautrelet et al. [START_REF] Gautrelet | Linearity investigation from a vibratory fatigue bench[END_REF] have determined the linearity range of the vibratory fatigue bench is 0-20g. According to this work, the linearity of the test system is studied from each response value at the frequency f r -5Hz with different acceleration levels. The values were extracted to plot the relationship between the response (strain or velocity CHAPTER 2. EXPERIMENTAL STRAIN-BASED VIBRATION CONTROL TO OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD response) and the excitation (acceleration levels). Here, f r is the resonant frequency, f r -5 is arbitrarily chosen before the resonance peak to keep the phase between the input and the output less than 20 • [START_REF] Česnik | Uninterrupted and accelerated vibrational fatigue testing with simultaneous monitoring of the natural frequency and damping[END_REF][START_REF] Gautrelet | Linearity investigation from a vibratory fatigue bench[END_REF]. Fig. 2.9 shows the relation between strain response and velocity response over the acceleration levels in resonant mode, and the linear curves are fitted by the least squares method. The fatigue test in this study is conducted within the linearity range of the test system.

Strain control strategy

Because of the inherent feature of the vibratory system, the HCF test on vibratory shaker is usually controlled by acceleration [START_REF] Cyprian | Comparative Analysis of Fatigue Energy Characteristics of S355J2 Steel Subjected to Multi-Axis Loads[END_REF][START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF][START_REF] Huertas | Resonant fatigue test bench for shaft testing[END_REF][START_REF] Mršnik | Frequency-domain methods for a vibration-fatigue-life estimation -Application to real data[END_REF][START_REF] Ogrinec | Vibration fatigue at half-sine impulse excitation in the time and frequency domains[END_REF]. By controlling the magnitude of the driving voltage, the acceleration of the vibration table is finally guaranteed to reach the specified value. This means the drive voltage of the vibration table can be adjusted slightly to keep the acceleration constant during the fatigue test. The strain measures obtained from acceleration control are dependent on the resonance amplification. Despite the use of identical specimens and the steady clamp, the dispersion in acceleration control leads to different strain amplitude values, so it is difficult to use in constant amplitude fatigue testing. That is why strain control is proposed in this study.

An experimental development on the shaker is conducted to define the strain level as an input control with a strain gauge placed on the specimen (see Fig. 2.10). In the left part of Fig. 2.10, the conventional vibration test on electro-dynamic shaker is usually controlled by the base acceleration (movements imposed on the clamping). Few recent controllers can be deflected to realize strain control.

On the contrary, the strain control in the right part makes sense in fatigue testing because the structure response is directly controlled. On the contrary, the acceleration and displacement depend on the FRF magnitude. The FRF of the test system shows variations with different specimens and clamping. These variations lead to large fluctuations of the stress or strain response amplitude, which may exceed 10% even for the same acceleration load. The fluctuation is too large to maintain stability, thereby reducing the reliability of vibration fatigue test results. Hence, strain control provides a constant stress or strain amplitude in the fatigue test. Besides, the vibration table works with the closed-loop control mode.

Experimental development for fatigue tests

In this study, the strain control fatigue test was implemented with the Sine Resonance Track-and-Dwell (SRTD) technique [START_REF] Ahmadi | Experimental high cycle fatigue testing and shape optimization of turbine blades[END_REF], which means the controller automatically maintains the resonance conditions during the test even though a change occurred (crack, for example). The SRTD test with strain control mode is illustrated in Fig. 2.11. This SRTD with strain control has two features. Firstly, the aim is to control the strain amplitude to keep a constant value. The shaker adjusts the base acceleration to reach the strain level needed for the fatigue test. Secondly, the SRTD maintains the phase difference between the excitation and response signals at 90 degrees. This ensures that the excitation frequency and resonant frequency of the specimen remain the same because a resonant frequency reduction is observed when a crack occurs. In general, the strain amplitude forms a closed-loop in SRTD test, which ensures a constant strain amplitude and working at the resonant frequency.

In the real test, the source signal for excitation is sent from the computer to the power amplifier and then driven the shaker to excite. The "Data Physics" software records the acceleration, velocity and strain signals in the frequency domain. The SRTD test makes it possible to record the evolution of the resonant frequency used in post-processing to define a criterion for assessing the presence of a crack on the The specimens were subjected to a sine excitation during the SRTD test, and the fatigue life under this strain amplification was subsequently identified. A schematic diagram of the data flow in the SRTD experiments is shown in Fig. 2.12. The source signals for excitation were sent from the computer to the power amplifier and then driven to the shaker to excite the specimens. The strain amplitude, initial resonant frequency, and the total number of cycles were defined at the beginning. The strain was controlled by a gauge and formed a closed-loop by the Data Acquisition (DAQ) system. The acceleration and velocity signals were also measured to obtain resonant frequency in real-time and calculate the number of cycles. The resonant frequency and the number of cycles were used to judge whether an SRTD test finishes. If both two criteria were not met, the resonant frequency was set as the excitation frequency for the amplifier to keep the test working at the resonant frequency. Otherwise, the SRTD test for one specimen was finished. All the signals of the acceleration, velocity, strain and excitation frequency were recorded in the time domain. The resonant frequency evolution recorded during the test was used in post-processing to define a criterion for evaluating the presence of cracks on the specimen.

According to Fig. 2.12, it is necessary to define the initial excitation frequency f r 0 , strain amplitude ε, and the fatigue cycles N L beforehand. The test system gives an excitation of the specimen to keep a constant strain response in resonant frequency. The system acquits phase shifts between input (strain) and output (acceleration) to obtain resonant frequency. The test maintains working frequency f r until 1 × 10 6 cycles, or stops due to instability caused by the critical crack.

With specimen DC01, the fatigue test procedures are depicted in Fig. 2.13. The test can be divided into two main steps for each specimen.



Step 2

Step 3 Note the desired strain amplitude by ε. For step 1, a sine sweep test with acceleration control (in base) is carried out from 190 Hz -220 Hz. The excitation amplitude is arbitrarily chosen with the aim that it leads to a strain response close to ε. This step has two objectives. Firstly it is necessary to determine the resonant frequency for the next step with strain control. Secondly, the FRF of this sub-test can be used OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD as an examination of the test machine and specimen to reduce the uncertainty of the test.

The resonant frequency, donated as f r ′ and obtained by step 1, is used for step 2. It should be noted that a much higher excitation voltage is required for the amplifier to achieve the same response for control in strain relative to control in acceleration. Since the response depends on modal characteristics in excitation on base, it is hard to reach high deformation far away from the resonant frequency. Consequently, the excitation frequency should be close enough to the resonant frequency to avoid sudden changes in excitation voltage. Upon many tests, the frequency range is set in (f r ′ -3, f r ′ +2), in Hz, for carrying out the sine sweep in strain on the low carbon steel specimens. Based on this frequency range, a sine sweep test with strain control in desire level is conducted to record the FRF corresponding to the ratio between the strain (control on the response) and acceleration (measure on base) channels. Thanks to this FRF, the related resonant frequency, denoted by f r 0 , is used as the initial excitation frequency in the SRTD test.

Step 3 is the main fatigue test carried with the SRTD test. The f r 0 from step 2 and ε are the input to SRTD test as shown in Fig. 2.12. The vibration excitation works at strain amplitude ε until the stop criterion.

Strain gauge and failure criterion

The fatigue damage of the strain sensor significantly affects the strain control strategy because both the specimen and the strain gauge are exposed to fatigue damage. Strain gauges are bonded to the center of the reduced zone to prevent premature gauge failure.

Empirically, the cracks were mainly observed on the face without the strain gauge. If the crack of the specimen leads to the fracture of a strain gauge, the specimen was not used for fatigue limit estimation to ensure all specimens reached the defined number of cycles.

Also, a threshold is required in the staircase method to define fatigue failure. The modal parameters of the mechanical system, such as resonant frequency and damping loss factor [START_REF] Česnik | Uninterrupted and accelerated vibrational fatigue testing with simultaneous monitoring of the natural frequency and damping[END_REF], shift when repetitive loading is applied to the system. However, Khalij et al. [START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF] have shown that the damping loss factor underestimates the number of cycles to failure. Therefore, the decrease in resonant frequency related to the crack presence and propagation was used in failure identification.

In order to avoid the different initial resonant frequency for each specimen, a normalized percentage rate changes between the resonant frequency of after (f r end ) and before (f r 0 ) SRTD test is used as s indicator defined as Change of Resonant Frequency (CRF), which is given by Eq. 2.1:

2.3. STRAIN CONTROL STRATEGY %CRF = f r 0 -f r end f r 0 × 100 (2.1)
To observe the strain gauge performance and related crack lengths to the thresholds, SRTD were carried out until CRF=2%, CRF=5% and CRF=8% of resonant frequency decrease. Fig. 2.14 reveals the relationship between frequency decrease and strain gauge measurement, as well as pictures obtained from a digital microscope (×300) of corresponding cracks on specimens. We can observe that:

(1) the strain keeps constant for a 2% decrease in frequency, and the cracks were not visible on the surface; (2) the strain level drops for a decrease in the resonant frequency of 5% but only 40 µm/m (which is about 3% to defined strain amplitude). The crack length roughly reached half-width of the reduction area (about 3.3 mm); (3) A dramatic drop of the strain value is observed after a 5% decrease in resonant frequency. The reason is that the crack on the surface of the specimen causes damage to the strain gauge. The crack extended over the whole width of the reduction area (6.8 mm) after an 8% frequency decrease.

In addition, the decrease in resonant frequency from 5% to 8% required a relatively small number of vibration cycles. Due to the severity of the crack, the strain gauge may lose its effectiveness after a decrease of 5%. Therefore, the stopping criterion for the staircase method is chosen at CRF = 5% which is also supported by the works of Gautrelet et al. [START_REF] Christophe | Resonance track-and-dwell testing for crack length measurement on 304L stainless steel[END_REF].

To sum up, the following characteristics need to be considered to ensure the accuracy and stability of the strain control:

(1) Make sure that the strain gauges of each specimen are bonded at the same position, which means the center of the reduced section. (2) Before the experiment, check whether the resistance value of the gauge is consistent with the nominal resistance value and balance the Wheatstone bridge before every test. (3) The fatigue limit of strain gauge HBM 1-LM11-1.5/350GE is 2000 µm/m in 1 × 10 7 cycles, which is much higher than that of the specimen. The cracks are always occurred in the specimen, and there is no fatigue crack in strain gauge in this study. (4) It must be noted that strain measurement is contact measurement, which means the effect of temperature evolution during the fatigue test should be examined first. Liakat et al. [START_REF] Liakat | On the anelasticity and fatigue fracture entropy in high-cycle metal fatigue[END_REF] deduced that the temperature of MCS 1018 carbon steel remains fairly unchanged in the HCF test, and the strain gauges are not sensitive to temperature during the room temperature range (20-30℃). Hence, the effect of temperature on the strain measurement of the fatigue test was considered negligible for DC01 steel.

Comparative study of fatigue test with strain control and acceleration control

The essential difference between acceleration control and strain control is that strain control realizes the control response during the fatigue process directly. In contrast, acceleration control is concerned with the input excitation of the specimen. The comparison between traditional acceleration control and proposed strain control is conducted on the sine sweep test and SRTD test to show the superiority of strain control.

STRAIN CONTROL STRATEGY

Sine sweep test

Firstly, a sine sweep test was conducted on one specimen to obtain dynamic characteristics of the specimen by acceleration control in 5 g and strain control in 1400 µm/m. These two amplitudes are chosen because of similar deformation responses. As shown in Fig. 2.15, the resonant frequencies obtained for these two control modes, 210.25 Hz and 210.60 Hz, are almost the same. The damping loss factors calculated from the half-power bandwidth method are similar: 1.132×10 -2 and 1.061×10 -2 , respectively. In addition, the phase information is displayed by calculating the phase 2.3. STRAIN CONTROL STRATEGY difference between the acceleration signal and strain signal. Strain control is a new control mode that leaves the modal characteristics of the mechanical system unchanged.

Sine Resonant Track-and-Dwell test

Secondly, the Sine Resonance Track-and-Dwell (SRTD) test for three specimens with 5 g acceleration control and for another three with 1400 µm/m, 1500 µm/m, and 1600 µm/m strain control were carried out. The strain amplitude and resonant frequency evolution of the total 6 specimens during SRTD test is compared in Fig. 2.16(a) and Fig. 2.16(b), respectively.

In Fig. 2.16(a), lines 1-3 represent the acceleration control, which is variable in both the initial and the decreasing progress of strain amplitude. Khalij's vibration fatigue test [START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF] revealed a reduction in the peak of FRF with a resonant frequency shift. This leads to two effects: the change in response amplitude during the fatigue test and an inconsistent response for different specimens. It is the reason for the variation in acceleration control. Moreover, the existent control ways in base, regardless of acceleration, velocity, or displacement, have an inherent disadvantage of diversity in response [START_REF] Česnik | Frequency-based structural modification for the case of base excitation[END_REF]. While lines 4-6, representing strain control, show a stable and constant strain amplitude during the test under different defined strain levels. The resonant frequency of the 6 specimens given in Fig. 2.16(b) differs in both acceleration control and strain control. The dynamic response of the structure varies a lot with different specimens due to different modal parameters. It is known that the resonant frequency decreases when a crack occurs, and this decrease continues with the crack propagation [START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF]. Though there are many variations, the strain control keeps a constant strain amplitude independent of the resonant frequency.

Combining Fig. 2.15 and Fig. 2.16, though the strain control has large fluctuations in amplitude compared to acceleration control, especially close to the resonant frequency in the sine sweep test, the strain control has a very stable response in the SRTD test. Hence, strain control is desirable for the purpose of obtaining a stable response for base vibration.

Microstructure observations

Additionally, the fracture surface of two specimens with acceleration control and strain control was observed to complete the comparison. The fatigue test results of these two specimens are listed in Tab. 2.5. The specimens were scanned in a high vacuum chamber of the ThermoFisher microscope. The specimens were positioned vertically on a glued holder and fracture surfaces were placed perpendicular to the 15 kV electron beam. Fig. 2.17 and Fig. 2.18 display Scanning Electron Microscopy (SEM) images of fatigue crack surface from strain control and acceleration control. acceleration control. Following the fatigue test, the specimen with strain control is submitted to tensile to fracture -to observe the fracture surface -because the vibration fatigue test does not necessarily lead to complete separation. The crack growth direction is indicated by yellow arrows. It is observed that the cracks are initiated only from the surface without strain gauge (points B and C) in strain control, while there is crack initiation on both surfaces (points A, B, C and D) in acceleration control. modes is that we observed many intergranular fractures in strain control, as shown in Fig. 2.18(a). The intergranular spread over the fracture surface, including crack initiation and propagation. Nevertheless, the intergranular fracture occurred but not so much as shown in Fig. 2.18(b). In accordance with conventional fatigue fracture of low carbon steel [START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF], the transgranular fracture is the predominant part of acceleration control. Indeed, the river-line patterns representative of transgranular fracture are most noticeable in this figure. In our opinion, this observation could be ascribed to different kinds of excitation and therefore different stress states between strain and acceleration controls. The centerline presented in Fig. 2.17(b) leads to the conclusion that the deformation is of the bending type, whereas Fig. 2.17

Strain gauge

Fatigue curve

To have a preliminary estimation of the fatigue limit, experimental tests with the strain control were carried out to obtain the ε -N fatigue curve, the corresponding Basquin's equation and the confidence. Specimens are cyclically loaded using sinusoidal signals at specified strain levels (ε) until the specimen reaches recognizable failure. SRTD described in Section 2.3.3.2 is used to excite the specimens. Thus, a number of ε -N points are plotted considering 5% decrease of the initial resonant frequency. Three strain amplitudes ε =1800 µm/m, 1500 µm/m, and 1200 µm/m are used as load levels, and three specimens are tested under each strain amplitude. The corresponding number of cycles to failure (N ) -considered as the fatigue life -is then extracted to plot the fatigue curve depicted in Fig. 2.19 by double logarithmic axis.

The ε -N equation is classically represented using the curve of the median fatigue data, but the variance information is then lost. It is necessary to construct a ε -N OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD curve that characterizes the confidence level and reliability, called the lower band curve, which ensures that the majority of the fatigue data falls above the lower bound value. The design curve can be derived by shifting the median ε-N curve, as expressed in Eq. 2.2.

Y P,C = Ŷ ± K × s (2.2)
where the subscripts R and C denote the reliability and confidence levels. Y and Ŷ donate the fatigue life (N ) in the lower band curve and median fatigue curve, respectively, s is the standard deviation of Y on fatigue strength. ± reflects the lower limit or upper limit with specific reliability and confidence level. To calculate the multiplier K, the double-sided confidence intervals approach and the approximate Owen one-side tolerance limit approach are introduced as follows. The double-sided confidence intervals approach [START_REF]Practice for Statistical Analysis of Linear or Linearized Stress-Life and Strain-Life Fatigue Data[END_REF][START_REF]Fatigue testing and analysis: theory and practice[END_REF] is used to obtain a confidence band with specific confidence given by:

K AST M = ± 2F C,(2,n-2) × 1 2 + (X i -X) 2 n k=1 (X i -X) 2 (2.3)
where F C,(2,n-2) is the F-distribution (also called Fisher-Snedecor distribution) value with the desired confidence interval C for (2, n -2) degrees of freedom. n is the test sample size. K AST M takes the negative value for the upper confidence band whereas the positive value for the lower confidence band. The approximate Owen one-side tolerance limit approach [START_REF] Shen | Design Curve to Characterize Fatigue Strength[END_REF] has been proposed to account for confidence and reliability.

K owen = K D × R owen K D = c 1 K R + K C c 3 K 2 R + c 2 a R owen = b 1 + b 2 f b 3 + b 4 exp(-f) (2.4) in which, K R = Φ -1 (R) K C = Φ -1 (C) f = n -2 a = 1.85 n (2.5)
where Φ is the standard normal Cumulative Density Function (CDF). The coefficients for empirical forms of K owen are shown in Tab. 2.6 and Tab. 2.7.

The double-sided confidence intervals provide a 95% confidence band, as presented in Fig. 2.19, accounting for the uncertainty of the median curve. The lower band curve is used for the design that ensures that there is a 95% possibility of survival with a 40 2.3. STRAIN CONTROL STRATEGY

(i = 1, 2, 3) for K owen Confidence level(C) c 1 c 2 c 3 f ⩽ 2 1 1 1 2f f > 2 1 + 3 4(f -1.042) f f -2 c 2 -c 2 1 Ref: Page 116 in [86]
90% of confidence level above this lower bound. The identification of the Basquin's parameters [START_REF] Basquin | The exponential law of endurance tests[END_REF] leads to:

ε = 1.255 × 10 4 × N -0.1606 (2.6)
Thus, the fatigue strength in 1 × 10 6 cycles -considered as fatigue limit in this study -is approached by the value of 1364.41 µm/m.

Application of staircase method for the fatigue limit estimation

This section describes the experimental staircase test with strain control. The test procedure and results are presented in this section.

Experimental procedure

The staircase test methodology [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] was extensively used to evaluate fatigue limit distribution by mean and standard deviation determination because of its simplicity and high accuracy. The specimens were tested sequentially, one at a time. The initial strain level ε 0 and strain step size d were determined before the test. The first specimen was tested at the initial strain level. If there is no crack (survival) for a pre-determined number of cycles N L (for limit value), the strain amplitude of the following specimen is incremented by one step size. Conversely, if the specimen cracks (failure), the strain amplitude for the next specimen is decremented by one step size. This manner is repeated in sequence, with the strain levels increasing and decreasing with the step size. The outcome is that approximately half of the specimens fail, and the other half do not.

The increment d has a great influence on standard deviation estimation. Zhao & Yang [START_REF] Zhao | Improved measurement on probabilistic fatigue limits/strengths by test data from staircase test method[END_REF] and Strzelecki & Sempruch [START_REF] Strzelecki | Experimental method for plotting S-N curve with a small number of specimens[END_REF], proposed the stress steps of 5 MPa and 6 MPa for a low carbon steel material. In accordance with these works, a strain step size of 20 µm/m is used in this study.

In this fatigue study, to carry out vibration tests, the 1 × 10 6 cycles are chosen for defining the fatigue limit. The first bending mode is excited because it gives the most severe damage compared to the other modes.

To summarize, the settings for the staircase tests are listed in the tab 2.8.

APPLICATION OF STAIRCASE METHOD FOR THE FATIGUE LIMIT ESTIMATION

With these set-up parameters, the test procedures for all specimens are schematically depicted in Fig. 2.20. In Fig. 2.13, each specimen was conducted with SRTD test (Section 2.3.3.2) until 1 × 10 6 cycles. After the fatigue test of one specimen, check the fatigue test results and determine the strain level for the next specimen. If the CRF was detected below 5% (survival) for 1 × 10 6 cycles, the strain level for the next specimen was incremented by one step size. Otherwise, it is decremented by a step size. After, the new strain level for another specimen is up to this test results. In the end, these steps were repeated on another until all specimens met the following three requirements for the staircase test. (1) Taking the first reverse results as a starting point of valid data, the number of valid specimens should be no less than 30 [START_REF]Metallic materials-Fatigue testing -Statistical planning and analysis of data[END_REF]. [START_REF] Shi | Approach and Application of Semi-Blind Source Separation for Aero-Engine Vibration Signals Using ICA-R[END_REF] The test contains at least three strain levels.

SRTD test (strain control)

(3) The strain level of the last specimen should be adjacent to the first strain level and have the opposite result [START_REF] Zhao | Probabilistic measurements of the fatigue limit data from a small sampling up-and-down test method[END_REF].

Test results and discussion

In order to obtain the fatigue limit of steel plates, a total of 36 valid specimens were conducted at staircase strain levels with the resonant frequency, donated as i from 1 to 36. According to the results, the up-and-down diagram of the staircase test is shown in Fig. 2.21. The ordinal number of failed specimens is listed in the figures. Fig. 2.22(a) shows the decreased resonant frequency. Six specimens failed due to CRF exceeding 5%, while the other four survived. Correspondingly, Fig. 2.22(b) depicted the strain levels at the preset value and kept them constant during the test. The disabling process of gauge, in the end, is speedy and occurs after the stop criterion so that the strain value could be seen as constant through the SRTD test, regardless of the variation in frequency. These data demonstrate the reliability of strain control in vibration fatigue test on electro-dynamic shaker. In addition, taking the evolution of the frequency shift of the No. 22 specimen as an example in Fig. 2.24, three stages can be distinguished in the progress of frequency decrease: adaptation phase, quasi-stationary phase and rapid failure phase. The decrease in resonant frequency was observed when the crack occured, and the decrease

APPLICATION OF STAIRCASE METHOD FOR THE FATIGUE LIMIT ESTIMATION

continues with the crack propagation [START_REF] Lorenzino | The variation of resonance frequency in fatigue tests as a tool for in-situ identification of crack initiation and propagation, and for the determination of cracked areas[END_REF]. Moreover, the 5% decrease chosen for the failure criterion is located in stage III.

variability of the damage

The detailed test results data of the 36 specimens, including the number of cycles with respect to different CRF and the number of cracks, are listed in Tab. 2.9.

It can be seen from the second column that the mass is not identical for all specimens, and the first mode resonant frequency may differ by a few Hertz due to the variability of the test system. However, these two parameters have no direct relationship with the result (survival or failure) and the number of cracks.

Because each of the SRTD test dose not stopped at the same number of cycles, the number of cycles and CRF at the end of the test is not the same for every specimen. Hence, the cracks at the end of the specimens are also variable. For example, Fig. 2.25 illustrates the number of cracks with different CRF. Fig. 2.25 (a) and (b) show that the number of cracks may be different in the same CRF, while Fig. 2.25 (c) and (d) show that the crack lengths are different with respect to CRF. The location of crack initiation and the length of crack propagation are variable in real fatigue experimental tests. However, cracks always occur on the side without the strain gauge.

To sum up, the strain control in a vibratory shaker for the fatigue test is validated by the staircase method. The estimation of fatigue limit from staircase results is provided in the following section.

Dixon-Mood method

As described in Chapter 1, the Dixon-Mood (DM) method [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] is commonly involved in estimating the fatigue limit distribution. The DM method was proposed by Dixon and Mood [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] in 1948, which provides approximate formulas of maximum likelihood estimation to calculate the mean and standard deviation under the Normal distribution.

In DM method, only less frequent events, failure or survival, are used to evaluate the distribution. The stress amplitude span is split by a step size d into several load levels numbered by j, where j = 0 stands for the lowest load level and j max stands for the maximum stress level. Denoting by n c,j the number of the fewer frequency events (survival or failure) at the load level j, two auxiliary values A and B can be calculated by Eq. 2.7: OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD

A = jmax j=0 j × n c,j B = jmax j=0 j 2 × n c,j n c = jmax j=0 n c,j (2.7)
The auxiliary values are used to estimate the median value m DM by Eq. 2.8 and the standard deviation s DM by Eq. 2.9. The minus sign given in Eq. 2.8 is used if the failed specimens are evaluated; otherwise, the plus sign is applied.

m DM = S a,0 + d A n c ± 1 2 (2.8) s DM = 1.62 × d Bn c -A 2 n 2 c + 0.029 if Bn c -A 2 n 2 c ⩾ 0.3 s DM = 0.53 × d if Bn c -A 2 n 2 c < 0.3 (2.9)
From up-and-down diagrams (Fig.

Lower limits of the fatigue limit

The rules of the staircase method make the test data concentrated near the median and distributed on both sides. Conducting fatigue tests at various strain levels in order to determine the median ε -N curve gives a probability of failure of 50%. With the DM, it is assumed that the variation of the fatigue limit follows a normal distribution. The one-sided tolerance limit method is used in this study to calculate the probability of fatigue strength.

In the fatigue life analysis, there is no problem if the real life is larger than life expectancy, but it cannot be too small. Similarly, the fatigue limit can be underestimated for safety, and it will be dangerous if the fatigue limit is overestimated. In this study, the one-sided confidence interval of a normal distribution is provided below.

After the statistical parameters (the mean µ and the standard deviation s) and the number of specimens n are available, the lower limit of the fatigue limit value for different confidence levels (C) and survival probability (P ) can be expressed as [START_REF]Metallic materials-Fatigue testing -Statistical planning and analysis of data[END_REF]: Low carbon steel is widely used in various building components, containers, furnaces, industrial machinery applications, etc. Thus, most research on the fatigue of steel has been carried out. In conventional stress-life domain, the database Papuga [START_REF] Papuga | FatLim -Database of Fatigue Limits[END_REF] gives the fatigue limit 268.7MPa of carbon steel for fully reverse bending. Delahay et al. [START_REF] Delahay | Estimation of the fatigue strength distribution in high-cycle multiaxial fatigue taking into account the stress-strain gradient effect[END_REF] gave a probability distribution of the fatigue strength at 2 × 10 6 cycles against the stress amplitude for C20 steel, as Fig. Though there are differences in load type and specimen geometry, an agreement between the results in this study and the literature data can be observed. While there is no standard test protocol for vibratory fatigue limit on the electro-dynamic shaker, this study provides a new experimental test and data processing method for estimating fatigue limit distribution.

µ P -C = µ -k(P, 1 -C, n)s (2.10) k = µ P + µ γ 1 n [1 -µγ 2 2(n-1) ] + µ P 2 2(n-1)

Conclusion

This chapter proposes a strain control test method applied on an electro-dynamic shaker to conduct a constant amplitude fatigue test. The efficiency of the proposed approach has been demonstrated by testing DC01 steel plates in the first mode. The strain amplitudes measured on the specimens remain constant and independent of the resonant frequency with the strain control, avoiding the diversity response of OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD traditional acceleration control in the base. The results highlight the repeatability of the strain-controlled test with the staircase method. The DM method provides an estimation of the fatigue threshold and the scatter.

The staircase approach provides a reasonable estimate of the median fatigue limit because about half of the specimens fail and the others do not [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF]. Nonetheless, it has a poor performance on the estimation of the standard deviation, especially for small sample tests [START_REF] Shi | Uncertainty evaluation by the bootstrap for the staircase fatigue limit test data[END_REF][START_REF] Klawonn | Efficient staircase testing of probabilistic Haigh diagrams[END_REF]. The results are: (1) It is more difficult to get an accurate measuring of dispersion due to the concentration of the data points near the median. (2) Usually, for the real fatigue tests, the fatigue limit distribution is hard to evaluate with a small number of specimens (n ⩽ 30). (3) Since the step size is chosen constant, the estimation of the fatigue limit is less accurate. Otherwise, a large number of specimens would be required to estimate the fatigue limit accurately. To avoid this, we propose, in the following chapter, to use bootstrap sampling to estimate the uncertainties. The next chapter presents a new statistical approach to improve the fatigue limit distribution estimation.

Chapter 3

Uncertainty quantification using resampling methods

In the last part of the previous chapter, the post-processing estimates the fatigue limit distribution from one experimental staircase test. However, the question arises as to whether the fatigue limit remains the same if the staircase test is conducted more than once under the same conditions.

In this chapter, the uncertainty analysis was carried out using the resampling method. A brief background of uncertainty for the staircase method is given in the first section. The second section details the resampling methods -leave-one-out and bootstrap -on the staircase data to evaluate the fatigue limit. Based on the experimental staircase test results of steel DC01, the uncertainty of the staircase test was evaluated by the bootstrap method. Then, the bootstrap was applied to the experimental data obtained for three kinds of specimens. The chapter ends with a brief conclusion. This part has been presented in an international congress [START_REF] Shi | Uncertainty evaluation by the bootstrap for the staircase fatigue limit test data[END_REF]. CHAPTER 3. UNCERTAINTY QUANTIFICATION USING RESAMPLING METHODS

Background

Since the fatigue limit is mainly estimated through experiments, uncertainties are consequently introduced into the assessment due to the inherent properties of the specimen as well as the environment of the experiment. These uncertainties may lead to an inaccurate estimation of the fatigue limit distribution. The uncertainty may originate owing to the fatigue limits of each specimen are random, and the fatigue process is affected in practice by many factors, such as the grain size, the type, the morphology and the loading [START_REF] Nadot | Fatigue from Defect: Influence of Size, Type, Position, Morphology and Loading[END_REF]. The measurement uncertainty of the test results also affects the fatigue test confidence. Hence, the assessment of measurement uncertainty is still required to evaluate the test results.

Therefore, it is commonly accepted that the fatigue design, including fatigue limit estimates, without considering uncertainties, may not be efficient. Pascual et al. [START_REF] Francis | Estimating Fatigue Curves with the Random Fatigue-Limit Model[END_REF] proposed a random fatigue-limit model to describe the variation in the fatigue limit and all fatigue curves. Rabb [START_REF] Rabb | Staircase testing -confidence and reliability[END_REF] analyzed the confidence level of outcomes of staircase and reliability of the standard deviation estimation based on the Monte-Carlo simulation (MCS). Wallin [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF] provided a study of the uncertainty of the Maximun Likelihood Estimation (MLE) method on staircase data. Jamalkhani Khameneh et al. [START_REF] Jamalkhani | Reliability prediction, scatter-band analysis and fatigue limit assessment of high-cycle fatigue properties in EN-GJS700-2 ductile cast iron[END_REF] deduced the mean, standard deviation and lower and upper scatter-band of the fatigue limit at different confidence levels from the staircase experiment. Karolczuk et al. [START_REF] Karolczuk | Fatigue life uncertainty prediction using the Monte Carlo and Latin hypercube sampling techniques under uniaxial and multiaxial cyclic loading[END_REF] used to predict fatigue life uncertainty by Monte Carlo and Latin Hypercube Sampling techniques.

However, fatigue limits can be expected to vary not only from specimen to specimen in one staircase test but also between different staircase tests. It is interesting to consider whether one staircase data can represent the real fatigue limit distribution. Hence, a question is what is the confidence level of one staircase test result.

Intending to evaluate the confidence of a fatigue limit distribution, this chapter describes the design and implementation of the resampling method on staircase test data. The resampling method, also used in statistical machine learning, is investigated in this paper as a possible means of assessing this uncertainty. By convention, no more than 30 specimens are loaded into an experimental staircase test for cost and time considerations. Resampling methods allow samples to be extracted repeatedly from the dataset and obtain additional information without more expensive tests.

This chapter discusses two resampling methods, including Leave-one-out crossvalidation (Loocv) and bootstrap. Loocv is a resampling method that splits all datasets one by one and computes a statistic with left-out one sample. The bootstrap method, developed by Efron [START_REF] Efron | Bootstrap Methods: Another Look at the Jackknife[END_REF], provides an efficient way to reproduce more samples from the small-size dataset. It is a reasonable way to estimate the epistemic uncertainty and provide non-parametric statistical inferences without any assumption about the 3.2. APPLICATION OF THE LEAVE-ONE-OUT RESAMPLING distribution describing the quantity under consideration. In a case that combined resampling and staircase test, Pollak et al. [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] used the bootstrapping algorithm to reduce the potential of sizeable standard deviation errors in the staircase tests. Leonetti et al. [START_REF] Leonetti | Uncertainty analysis of constant amplitude fatigue test data employing the six parameters random fatigue limit model[END_REF] employed the Bootstrap method to linear regression of the S-N curve. A literature review indicated that almost no such resampling method had been mentioned for the staircase test uncertainty evaluation. We focused on the bootstrap method analysis, which allows the uncertainty assessment of the fatigue limit based on only existing test data. In this study, the bootstrap to extract more information from the limited experimental data is applied initially to assess the uncertainty of the staircase test data.

Application of the leave-one-out resampling

Leave-one-out cross-validation (Loocv) is a typical cross-validation method in machine learning, especially for small samples conditions. Cross-validation means that the dataset is randomly divided into k parts, of which there are k -1 parts in the training set and 1 part in the test set. The first data group can be used to test the model fit based on the remaining k -1 pieces of data, and then the second data group can be used as a test set to examine the adequacy of the corresponding training set.

It is commonly known that the larger the number of specimens, the more accurate the staircase data can represent the true fatigue limit distribution. In the literature of the last decades, there is almost no real experimental staircase test that exceeds 30 specimens. Inspired by the Loocv method, a similar Leave-one-out (Loo) resampling method can be used to measure the number of specimens. For the staircase test, every single piece of data represents an observation of one specimen in the real test. Therefore, each strain (or stress) value can be regarded as a part, and all test data is divided into n parts, where n is the total number of specimens in a staircase test. Assuming that N = n -1 samples are taken out in each sampling to realize, leave one sample is left out. Using all staircase load (e.g. stress or strain amplitude) data as the dataset, the general procedure is according to Algorithm 3.1. As a resampling method, Loo has two main advantages:

(1) It is much less biased. We use a repeated sampling set containing n -1 observations, which is almost identical to all observations in the entire data set. Therefore, the Loo method will not lead to other data errors like other cross-validation methods.

(2) It is not computationally expensive to perform the Loo procedure. METHODS Algorithm 3.1: Leave-one-out resampling [START_REF] William | Mechanical Behavior of Materials[END_REF] The staircase dataset that including n specimens is represented as The analysis of the Loo method is performed to determine the efficiency of a single specimen. In this analysis, the number of samples N is chosen as N = n-1 = 35 in the numerical test, and a total of 36 times resampling (M ) is used. With Loo resampling method, the distributions from all resampling data are shown in Fig. 3.1. Each line in Fig. 3.1 provides one Loo resampled result. The Loo excludes one data (tested strain level) at each sampling, so 36 samplings were repeated in this case. However, the Loo sampling for eliminating the same tested strain value leads to the same result. The maximum difference between the estimated median and standard

X = [x 1 , x 2 , ..., x n ] 2 Let N = n -1,

BOOTSTRAP RESAMPLING AND RESULTS

deviation between the experimental data and resampled data is 0.096% and -5.26%, respectively. It means that the number of specimens, n = 36, is sufficient in this experimental test. The test with one more specimen will change the result slightly.

Bootstrap resampling and results

Procedure

The fatigue limit is regarded as a random variable, while the staircase is an experimental method to estimate the distribution of this variable. Bootstrap provides a non-parametric method that relies on resampling the observed samples. That is, it uses several empirical distributions to approximate the population distribution. This method fully uses the given observation information without other assumptions of the numerical model and adds any new observations. The critical point of the bootstrap is unordered sampling with replacement. Combined with the staircase method, a bootstrap procedure is proposed in Algorithm 3.2. (N is the number of specimens in a single bootstrap sampling, and M is the number of bootstrap replications).

In order to evaluate the resampling results, the uncertainty factors of the median and standard deviation, δ m and δ s , are defined as:

δ m = m -m ex m ex × 100% δ s = s -s ex s ex × 100% (3.1)
where m and s are the median and standard deviation, subscript "ex" is the value from experimental data. This bootstrap procedure utilizes multiple samples randomly drawn from experimental test results to make a statistical evaluation of the primary dataset. Several advantages of using bootstrap on the staircase data can be noted. Firstly, compared with the Loo method, the bootstrap can avoid the problem of reducing the data by resampling. Secondly, the bootstrap can also create randomness in the data, which prevents the time-consuming repeated real staircase tests.

Experiment-based hyper-parameter sensitivity analysis

As described in Algorithm 3.2, there are two parameters M and N in the bootstrap method, where N is the number of specimens in a single sampling and M is the number of bootstrap replications. A question is how to find a suitable value for M and N with METHODS Algorithm 3.2: Bootstrap 1 The staircase dataset including n simples is represented as X = [x 1 , x 2 , ..., x n ], and the step size is d 2 Define N as the number of samples in a single sampling and M as the number of bootstrap replications 3 Find all strain (or stress) levels L = [l 1 , l 2 , . . . l] 4 Divide X to different part according to different strain (or stress) level l, let X l = [x l 1 , x l 2 , ..., x l ] , that is X 1 ∪ X 2 ∪ ...X l = X 5 Pre-define: select randomly a test strain level l 6 get all test data in strain level l 7 foreach i in M do

Results and discussion

The bootstrap was performed 100 times with 100 samples in each sampling. The results are shown in Fig. 3.4. The light blue lines are all bootstrap resampling results, the blue dash represents an average Probability Density Function (PDF) of all sampled data, and the red dash represents PDF of the experimental data. median and standard deviation of the experimental data. The black dashes represent 5 th percentile and 95 th percentile. We can see that the range of 5 th percentile and 95 th percentile is no more than 3% for the median value, but it is almost more than 30% for the standard deviation value. The uncertainty for the standard deviation estimation is substantial, and much higher than the median estimation.

Effect of specimen dimensions

Two other kinds of specimens are used in this section to investigate the frequency and thickness impacts on the fatigue limit uncertainties. The results are also analyzed by bootstrap sampling.

Staircase tests

Three kinds of specimens, including the one described in Chapter 2, have the same specimen shape and reduced section, as shown in Fig. 3.6. The structural difference in thickness and length affects the resonant frequency. The resonant frequencies are obtained from the experimental sine-sweep test of each kind of specimen. The differences in thickness, length and resonant frequency are listed in Tab. 3.1. The material (steel DC01), manufacture, surface treatment and experimental environment are identical for these specimens. Using the experimental setup and test METHODS procedures described in Section 2.4, tests on the other two specimens are conducted with strain control. The same step size d=20 µm/m for the staircase method. The test results are detailed in Appendix B, including frequency, cracks, and so on. The up-and-down diagrams for specimens S1 and S2L are shown in Fig. 3.7 and Fig. 3.8, respectively. Similarly to Section 2.4.2.3, the DM method is used to estimate the distribution of staircase results, as shown in Fig. 3.9 and Fig. 3.10. These figures show a difference in fatigue limit between the three kinds of specimens. The estimated distributions of S1, S2 and S2L are represented by orange, blue and pink lines, respectively. The cross marks used in Fig. 3.10 represent the probability of failure for each test strain amplitude.

The means and standard deviations are reported in Tab. 3.2. The fatigue limit medians are different. Indeed, the fatigue limit of S1 is 50.84 µm/m higher than the S2 value, and fatigue limit of S2 is 54.46 µm/m higher than S2L value. Less difference in the estimated standard deviation is observed.

Application of the bootstrap sampling

Choosing N = 100 and M = 100, the bootstrap results obtained for the three kinds of specimens are shown in Fig. 3.11. The thick lines represent the distribution estimated from the experimental data by DM method. The thin lines represent distributions estimated from the bootstrap data. In order to compare the overlapping of the three clusters of PDF, the median values of all sampled results are marked as pentagon, star and circle, respectively. From this figure, we can see that there is no overlap between the three PDFs. We relied on the literature. In this study, the specimens have two main differences: loading frequency (due to the change in the length of the specimens) and geometry dimensions, such as the thickness of the cross-section and the length. Based on all the resampled results in Fig. 3.11, the distributions of all these median and standard deviation values from different specimens are shown in Fig. 3.12 and Fig. 3.13.

In Fig. 3.12 and Fig. 3.13, the vertical red dashes represent the estimated median and standard deviation of the experimental data, other vertical dashes represent 5 th percentile and 95 th percentile of each kind of specimen. Similar to the observations METHODS Figure 3.13: Distributions of standard deviation from the bootstrap results of S1, S2, and S2L from Fig. 3.5, the uncertainty for the standard deviation estimation is much higher than the median estimation. From Fig. 3.13, it can be found that the bootstrap results always have a more significant deflection on the left of the experiment result (red dash). It is proof that the estimation of the standard deviation is systematically less accurate and lower than the true value [START_REF] Randall D Pollak | Analysis of Methods for Determining High Cycle Fatigue Strength of a Material With Investigation of Ti-6Al-4V Gigacycle Fatigue Behavior[END_REF].

Regarding frequency effect, Tsutsumi et al. [START_REF] Tsutsumi | Effect of test frequency on fatigue strength of low carbon steel[END_REF] highlighted the frequency effect by comparing the fatigue tests at 20 kHz and 10 Hz. The stress-strain curves also highlighted this effect at a relatively high strain rate by Khalij et al. [START_REF] Khalij | Fatigue curves of a low carbon steel obtained from vibration experiments with an electrodynamic shaker[END_REF]. However, the excitation frequency (when less than 200Hz) has little effect on the fatigue limit, according to the research of Dallali et al. [START_REF] Dallali | Effect of geometric size deviation induced by machining on the vibration fatigue behavior of Ti-6Al-4V[END_REF] and Tsutsumi et al. [START_REF] Tsutsumi | Effect of test frequency on fatigue strength of low carbon steel[END_REF].

In our opinion, it is the combination of several parameters that affect the fatigue limit, especially since the tests were carried out in vibration. The biggest difference here is seen when a change in length and thickness is combined (comparison between S1 and S2L). It could therefore highlight the effect of bending stress gradient. Indeed, Narumoto et al. [START_REF] Narumoto | Effect of Plate Thickness on the Fatigue Strength of Cruciform Welded Joints[END_REF] considered that the difference in bending stress gradient is one of the causes of the size effect. They also showed that the stress intensity factor increases with thickness, resulting in faster crack growth. Narumoto et al. [START_REF] Narumoto | Effect of Plate Thickness on the Fatigue Strength of Cruciform Welded Joints[END_REF] and Bhuyan et al. [START_REF] Bhuyan | Effect of thickness on fatigue behaviour of notched plates[END_REF] showed that the fatigue strength decreases with increasing thickness. However, the authors studied plates with greater thicknesses. In another work, Bhuyan et al. [START_REF] Bhuyan | Effect of thickness on fatigue behaviour of notched plates[END_REF] found that the proportion of total fatigue life spent in crack initiation increases with the thickness decrease. They pointed out that the strain rate, induced by the thermally activated movement of dislocations in the ferritic microstructure of low carbon steel, has an effect on the fatigue limit.

Conclusion

This part presents an evaluation of statistical uncertainty in the staircase test for fatigue limit based on resampling methods. To avoid repetition of the real staircase tests, a Loo and the bootstrap procedure is used to produce more artificial staircase data based on one staircase test. Experimental staircase data from the vibration bending fatigue test on DC01 steel specimens were analyzed.

From the resampling results, the obtained conclusions could be listed as follows:

(1) Loo method can be used to judge if the number of specimens is sufficient in a staircase test. (2) To get a steady result, it is necessary to use N ⩾ 100 specimens and conduct M ⩾ 100 times of staircase test.

(3) The results from bootstrap resampling show that the standard deviation of fatigue limit is more dispersive with more than 20%, leading to a conclusion that it should be conservative to use staircase results in fatigue design.

However, variability in the fatigue experiment test is unavoidable. Several main factors are involved in the scattering of fatigue test results. Some of these have been addressed by Murakami et al. [START_REF] Murakami | Essential structure of S-N curve: Prediction of fatigue life and fatigue limit of defective materials and nature of scatter[END_REF]. Due to this considerable uncertainty in standard deviation estimation, the improvement to the staircase method from post-processing is conducted in the next chapter.

Chapter 4

A non-parametric evaluation method in staircase test for improving fatigue limit assessment

The previous works provided experimental staircase test results and the uncertainties of the result. The estimation of uncertainties cannot be obtained directly by experimental tests because it requires many specimens and time. Under this, a numerical post-processing method is needed. In this chapter, a new evaluation method based on Kernel Density Estimation (KDE) is proposed to estimate the fatigue limit distribution from staircase tests without prior knowledge. This chapter aims to summarize the methods, improve the estimation performance, and validate KDE effectiveness by comparing it to other evaluation techniques.

The first section presents a brief survey of existing statistical methods for staircase data. Then a novel non-parametric method was proposed based on KDE. A simulation-based investigation was performed to compare the effectiveness of the proposed method to other evaluation methods in the following section. The obtained results in the third section reveal that the proposed technique offers a better estimate of the median, the standard deviation and the probability distribution of the fatigue limit. The non-parametric method applied to experimental data is presented in the fourth section. The work of this part has been published in Probabilistic Engineering Mechanics [START_REF] Shi | A non-parametric evaluation method in staircase test for improving fatigue limit assessment[END_REF].

. Existing evaluation techniques

As described in Chapter 1, the staircase test method [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] is commonly involved in characterizing the fatigue limit distribution. However, several researchers have investigated ways to improve the accuracy of the fatigue limit estimation by improving evaluation techniques in post-processing. Müller et al. [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] compared several evaluation techniques through Monte-Carlo simulation for evaluating experimental results. However, the fatigue scatter is still strongly dependent on step size. This background is introduced in detail in following.

The inadequacy of standard deviation estimation on staircase tests has received considerable attention in the literature. Several evaluations techniques [START_REF] Lin | Evaluation of the staircase and the accelerated test methods for fatigue limit distributions[END_REF][START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF][START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF][START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF][START_REF] Braitner | Use of the parallel-projected and staircase method to predict fatigue strength of ASTM A743 CA6NM alloy steel[END_REF] have been proposed since the first development of the staircase method. Tab. 4.1 highlights some notable methods in recent years. As the author of the staircase test method, Dixon et al. [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] proposed the Dixon-Mood (DM) method based on Maximun Likelihood Estimation (MLE). However, several studies reveal that this evaluation technique gives a reasonable estimation of the mean value but underestimates the standard deviation of the fatigue limit. To address this bias, some researchers enlarged the DM estimation on the standard deviation by correcting factors. Svensson [START_REF] Svensson | Random Features of the Fatigue Limit[END_REF] proposed a correction function with number of specimens as variable, Braam et al. [START_REF] Jj Braam | A Statistical Evaluation of the Staircase and the ArcSinP Methods for Determining the Fatigue Limit[END_REF] used a correction function with number of specimens and step size but has been proved worse estimated with small sample size. Pollak et al. [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] improved the estimation by a non-linear function. For tests with a small number of specimens, Pollak [START_REF] Randall D Pollak | Analysis of Methods for Determining High Cycle Fatigue Strength of a Material With Investigation of Ti-6Al-4V Gigacycle Fatigue Behavior[END_REF] suggested combining the bootstrap sampling technique with correcting factors to get a better estimate.

An alternative usage of MLE in post-processing staircase data is to use the censored data [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF][START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF][START_REF] Bai | Measurement and estimation of probabilistic fatigue limits using Monte-Carlo simulations[END_REF] with or without a correcting function on the standard deviation bias. Meanwhile, other research has introduced statistical techniques into the staircase data processing. Wallin [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF] proposed an evaluation method based on binomial probability to analyze staircase data, whereas the method yields an acceptable estimation only when the staircase test step size is close to the real value of the standard deviation. Zhao et al. [START_REF] Zhao | Improved measurement on probabilistic fatigue limits/strengths by test data from staircase test method[END_REF] developed a general maximum likelihood approach (GMLA) to assess the fatigue limit by constructing physically paired local S-N relations for all failure or survival specimens from staircase tests. The IABG (Industrieanlagen-Betriebsgesellschaft) method [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] provides an estimator by omitting invalid test results and adding fictitious data. From the simulation study [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF], the differences between the IABG and MLE methods are negligible. Other methods such as Zhang-Kececioglu method (ZKA) [START_REF] Zhang | New approaches to determine the endurance strength distribution[END_REF] have no inherent improvement in standard deviation estimation and have been proven to perform less than MLE [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF][START_REF] Zhao | Probabilistic measurements of the fatigue limit data from a small sampling up-and-down test method[END_REF]. Deubelbeiss' method is deduced from the regression of the failure probability, but leads to a significant bias on the standard deviation [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF].

To sum up, firstly, most of the proposed statistical analysis methods are based on the MLE. Therefore, these methods have to make assumptions about the underlying distribution ineluctably. Secondly, these evaluation techniques lead to different accuracy, more or less correct. However, the drawback of the strong dependence on the step size still exists. Among all these kinds of evaluation techniques, the MLE and the DM method are generally accepted by researchers. Thus, these two methods are used as references in this work. The simulation on DM and MLE are illustrated in Appendix C and Appendix D respectively.

Dixon-Mood method and corrections

The DM method was proposed by Dixon and Mood [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] in 1948, which provides approximate formulas of maximum likelihood to estimate the mean and the standard deviation. It assumes that the fatigue limit should follow the Normal distribution. Only the less frequent events, failure or survival, are used to evaluate the distribution. The stress amplitude span is split by a step size d into several load levels numbered by j, where j = 0 stands for the lowest load level and j max stands for the maximum stress level. Denoting by n c,j the number of the fewer frequency events (survival or failure) at the load level j, two auxiliary values A and B can be calculated by Eq. 4.1:

A = jmax j=0 j × n c,j B = jmax j=0 j 2 × n c,j n c = jmax j=0 n c,j (4.1)
The auxiliary values are used to estimate the median value m DM with Eq. 4.2 and the standard deviation s DM with Eq. 4.3. The minus sign given in Eq. 4.2 is used if the failed specimens are evaluated and otherwise, the plus sign is applied.

m DM = S a,0 + d A n c ± 1 2 (4.2) s DM = 1.62 × d Bn c -A 2 n 2 c + 0.029 if Bn c -A 2 n 2 c ⩾ 0.3 s DM = 0.53 × d if Bn c -A 2 n 2 c < 0.3 (4.3)
As listed in Tab. 4.1, some corrections to the DM method have been proposed to reduce the effect of the step size. The simulation on DM in Appendix C proved that the correction factor (Eq. 4.4) proposed by Svensson-Lorén (SL) [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] gives a better improvement to the standard deviation estimation:

s SL = n n -3 × s DM (4.4)
Therefore, the SL correction is used as a reference method in this simulation-based investigation.

NON-PARAMETRIC EVALUATION METHOD AND SIMULATION PROCEDURE

Maximum Likelihood Estimation method

The Maximun Likelihood Estimation (MLE) provides an appropriate tool for solving the general problem of estimating the "best fit" line through censored test data. Thus, the resulting estimates are those that agree most closely with the observed data. MLE gives a more precise estimate than any other methods, and is adaptive. It can be used for any kind of method with any type of distribution. Its main drawback is that the likelihood function has a different form for each specific distribution.

The purpose of the MLE method is to use the known sample results to infer the most likely, in other words, the maximum probability parameter values that lead to such results. When the stress amplitude S a of the staircase test is known, the failed and survived specimens give information about the probability:

P(x > S a ) = 1 -F (S a ) P(x ⩽ S a ) = F (S a ) (4.5)
where F (•) denotes the Cumulative Density Function (CDF).

Let n f,j be the number of failures and n s,j be the number of survivals at the stress level S a,j , the maximum likelihood function L is given by:

L = jmax j=0 F (S a,j ) n f,j (1 -F (S a,j )) n s,j (4.6)
The maximization of L results in the estimation of probability distribution parameters. Although an assumption about the underlying distribution is required when performing the MLE method, this method can deal with the non-Normal distribution of the fatigue limit and is also widely used in the literature [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF]. The simulation study of MLE method is presented in Appendix D.

Non-parametric evaluation method and simulation procedure

In order to improve the standard deviation estimation on fatigue limit for small sample tests, an evaluation technique based on Kernel Density Estimation (KDE) and non-linear correction is proposed in this work to post-process staircase tests. The KDE was applied to estimate the fatigue limit from experimental staircase tests by the authors [START_REF] Shi | Experimental strain-based vibration control to obtain the fatigue strain limit by the staircase method[END_REF]. The current work focuses majorly on:

-Implementation of a parameter-free estimator resulting in an accurate estimation of both the median and the standard deviation of the fatigue limit. TEST FOR IMPROVING FATIGUE LIMIT ASSESSMENT -Effectiveness of the proposed method is less dependent on the test parameters from the staircase method, especially the step size. -Probabilistic fatigue limit estimation without prior knowledge of the fatigue limit distribution.

Non-parametric fatigue limit estimation

As a non-parametric method to estimate the PDF of random variables, KDE has been proven to be suitable for post-process experiment data like staircase tests [START_REF] Bai | Machine learning assisted probabilistic prediction of long-term fatigue damage and vibration reduction of wind turbine tower using active damping system[END_REF]. Compared to other post-processing methods reviewed in the previous section, the KDE method has the advantage of being non-parametric and non-required on the prior knowledge of the underlying distribution. In the case of staircase tests, Dixon [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF] pointed out that the load levels fluctuate around the median value of the fatigue limit. The fatigue limit can be considered as a random variable. Based on the staircase experiment rules, the ratio between the number of specimens at each load level and the total number of specimens represents the probability of the fatigue limit at this load level. The greater the number of specimens is, the higher the probability of the fatigue limit occurring, i.e., (Eq. 4.7):

P {x = S f } ∝ n j n (4.7)
where, S f is the fatigue limit, n j is the number of specimens at the load level j and n is the total number of specimens (or samples). Let a random data set X = {x 1 , x 2 , ..., x n } in the probability space χ, denote all the load amplitudes in the staircase test, the PDF of the fatigue limit distribution estimated by KDE is:

fh (x) = 1 nh n i=1 K x -x i h (4.8)
where, h > 0 is a smoothing parameter called bandwidth, K is the non-negative kernel function using the standard Gaussian Kernel [START_REF] Plesovskaya | An empirical analysis of KDEbased generative models on small datasets[END_REF] defined as:

K(u) = 1 √ 2π exp - u 2 2 (4.9)
Eq. 4.9 is applied to each sample from staircase tests. The PDF of all samples is then estimated by the sum of these kernel densities on every data point. The KDE estimator does not distinguish between specimens with and without failure. To optimize the performance of KDE in fatigue limit estimation, we compared different bandwidth (h) methods of KDE as described in the next section.

NON-PARAMETRIC EVALUATION METHOD AND SIMULATION PROCEDURE

Bandwidth selection

In KDE method, a smoothing parameter (a.k.a. bandwidth h) is required to control the smoothing scale and the density estimation of the underlying distribution [START_REF] Rosenblatt | Remarks on Some Nonparametric Estimates of a Density Function[END_REF]. Several methods have been proposed to optimally select this hyperparameter [START_REF] Heidenreich | Bandwidth selection for kernel density estimation: a review of fully automatic selectors[END_REF][START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF].

If the Gaussian kernel function is used to approximate data for an underlying Gaussian distribution, Scott's rule (Eq. 4.10) and Silverman's rule (Eq. 4.11) are two efficient ways:

h SC = 4s 5 3n 1/5 ≈ 1.06 × sn -1/5 (4.10) h SI = 0.9 × min(s, IQR/1.35)n -1/5 (4.11)
where, n is the number of samples, s is the standard deviation of the data and IQR is the interquartile range. In addition, some research applied the Sheather-Jones method which reintroduces a non-stochastic term to reduce bias in estimation without inflating variance [START_REF] Simon | A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation[END_REF]. The Sheather-Jones method is more suitable for processing multimodality distribution. These data-based methods give a variable bandwidth and it is challenging to delimit whether it fits the staircase data. Consider the DC01 staircase experimental data (in Section 2) as an example, the estimated PDF from SC (Scott), SI (Silverman), SJ (Sheather-Jones) bandwidths are shown in Fig. 4 With the left y-axis in Fig. 4.1, the orange bars represent staircase experimental data at five strain levels. The right y-axis corresponds to the estimated PDF from SC, SI, SJ bandwidths, with the value of the bandwidth listed in the right legend. It should be noted that the fatigue limit distribution should be subject to flat unimodal distribution because we only consider one mode of failure. Nevertheless, these four kinds of bandwidth all give non-smooth distributions. Comparatively, SC uses the largest bandwidth and only one peak in PDF. Conversely, SJ method uses the smallest bandwidth and produces extreme multi-peaks PDF curve. SI and Normal-reference method also leads to obvious multipeaked distribution. SC is only one bandwidth selector that leads to one peaked distribution but still has signs of multimodality. In order to make the estimated distribution consistent with the physical meaning of fatigue, this study try to enlarge the coefficient in SC equation (Eq. 4.10) based on MCS. For optimizing the bandwidth, the corrected equation based on SC is shown as:

h opt = α × 4s 5 3n 1 5 (4.12)
where α is the coefficient that adapted to Scott's rule. The simulation study on how to adjust the parameter is presented in Section 4.

Bias correction on fatigue limit standard deviation

Since it is known that linear correction enlarges the estimation uncertainty [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF], this study developed a nonlinear approach to reduce the standard deviation bias as follows:

   s KDE = s d + As c if s > d s KDE = s d + Bs c if s ⩽ d (4.13)
where, s is the standard deviation from KDE with optimized bandwidth, s KDE is the corrected standard deviation. This study proposes this formula based on correction equations in the literature. Some of them are introduced and simulated in Appendix C. The coefficients A, B and c, shown in Tab. 4.2, are correction coefficients depending on the coefficient of variation (CV). Tab. 4.2 was determined by simulation tests as described in the following Section 4.3.1. In order to correct the estimated standard deviation s close to the initial true standard deviation, we carried out lots of tests to determine the coefficients A, B, The proposed KDE-based fatigue limit estimation with bias correction is expressed by the pseudo-code in Algorithm 4.1. In order to evaluate the effectiveness of the proposed method, a numerical simulation modeling the staircase test was practiced. The staircase simulation is configured as shown in Fig. 4.3. Similar simulation procedure and result evaluation to Reference [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] were conducted in this work, but more distributions and coefficients of variation were investigated. Firstly, a simulated fatigue limit distribution was initialized by a distribution model and a coefficient of variation (CV) selected by the authors. Since the mean value µ 0 was considered constant in this study, CV controls the standard deviation of the fatigue limit s 0 directly. The simulated fatigue limit distribution is then generated by the mean value µ 0 and the standard deviation s 0 with different distribution models, including Normal, Lognormal and Weibull distributions. In this step, the mean is arbitrarily fixed at µ 0 = 400, and the standard deviation is controlled by CV (CV=0.0125, 0.025, 0.05). These three values selected in this study come from Roué et al. [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] and previous experimental data. Then, the distribution was constructed by mean (µ 0 = 400) and standard deviation (s 0 = CV × µ 0 ). The shape, location and scale parameters were calculated from the mean and standard deviation. However, as pointed out by Müller et al. [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] and Roué et al. [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF], the fatigue limit estimated from the staircase test represents the median threshold with a failure probability of 50%. Consequently, the median value and the standard deviation were used in this work to describe the fatigue limit distribution. The mean value (µ) and the standard deviation (s) can be calculated by the discrete PDF as Eq. 4.15. The median (m) can be obtained from the discrete CDF with cumulative probability 50%.

µ = max i=0 x i P (x i ) s 2 = max i=0 (x i -µ) 2 P (x i ) max i=0 P (x i ) (4. 15
)
where i is ordinal number, x i is random variable, P (x i ) is the probability density of variable x i . Secondly, in the numerical simulation, the specimens were generated successively with respect to the staircase method. For the i th specimen, a value (a.k.a. fatigue limit of i th specimen) was randomly extracted from the fatigue limit distribution set in the initialization step. All random processes involved in the work are practiced by "scipy.stats.rv_continuous" class from SciPy package [START_REF] Virtanen | SciPy 1.0: fundamental algorithms for scientific computing in Python[END_REF]. This value was then compared to the load level (a.k.a. applied stress amplitude) of the i th specimen. Just as in experimental tests, the specimen is donated as "survival" if the applied stress amplitude is below the fatigue limit, otherwise, the specimen is donated as "failure". The survival/failure state of the i th specimen determines the stress amplitude that will be applied to the (i + 1) th specimen. Then, the step size between load levels is fixed to a constant d. In other words, if the i th specimen is survival at the load level S i a , the stress amplitude S i+1 a for the (i + 1) th specimen equals to S i a + d. Contrarily, if the i th specimen is failure, the stress amplitude S i+1 a for the (i + 1) th specimen equals to S i a -d. As for the initial specimen (i.e., i = 1), the load level S i=1 a is set at the mean fatigue limit µ 0 of the given distribution. The above steps are repeated until enough specimens of n are generated. With the consideration of reducing the effect of starting load level, n is the nominal sample size [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF][START_REF] Zhao | Probabilistic measurements of the fatigue limit data from a small sampling up-and-down test method[END_REF] In order to evaluate the effectiveness of post-processing methods in fatigue limit estimation, five cases were constructed by combining a probability distribution with a CV. For each study case, a total number of N = 1000 trials [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] were performed to investigate the quality of the estimates with respect to the number of specimens n ∈ 10, [START_REF] Endo | A practical method for fatigue limit prediction in ductile cast irons[END_REF][START_REF] Svensson | Random Features of the Fatigue Limit[END_REF][START_REF] Cetin | The fatigue limit: An analytical solution to a Monte Carlo problem[END_REF][START_REF] Boresi | An appraisal of the prot method of fatgiue testing Part I[END_REF][START_REF]Metallic materials-Fatigue testing -Statistical planning and analysis of data[END_REF][START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF][START_REF] Nicholas | High cycle fatigue: a mechanics of materials perspective[END_REF][START_REF] Presas | Fatigue life estimation of Francis turbines based on experimental strain measurements: Review of the actual data and future trends[END_REF][START_REF] Dallali | Effect of geometric size deviation induced by machining on the vibration fatigue behavior of Ti-6Al-4V[END_REF] and the normalised step size d ∈ 0.1, 0.2, • • • , 2 listed in Tab. 4.3. The evaluation methods and simulation variables are to be assigned to each case. In most researches, the fatigue limit distribution is assumed to be the Normal distribution by default [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF][START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF]. However, the probability distribution shape may have little effect on the fatigue limit mean estimate but its tail is important for estimating

NUMERICAL VALIDATION OF THE PROPOSED EVALUATION METHOD

the dispersion [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF]. Some recent studies have revealed that the distribution shape of the fatigue limit is not symmetric [START_REF] Zhao | An approach for determining an appropriate assumed distribution of fatigue life under limited data[END_REF] and have suggested the usage of Lognormal distribution [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF][START_REF] Rabb | Staircase testing -confidence and reliability[END_REF][START_REF] Bai | Measurement and estimation of probabilistic fatigue limits using Monte-Carlo simulations[END_REF]. Moreover, the Weibull distribution was also proved to be a suitable estimation for the fatigue limit standard deviation [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF][START_REF] Çalişkan | Determining the endurance limit of AISI 4340 steels in terms of different statistical approaches[END_REF][START_REF] Strzelecki | Application of Weibull distribution to describe S-N curve with using small number specimens[END_REF]. Hence, these three probability distributions (i.e., Normal, Lognormal, and Weibull) are applied in this work to study the usability of the proposed KDE-based evaluation method. The CV value close to 0.0125 obtained in the experimental staircase tests carried out by Shi et al. [START_REF] Shi | Experimental strain-based vibration control to obtain the fatigue strain limit by the staircase method[END_REF] is used in this work. Meanwhile, Roué et al. [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] proposed to investigate the CV of 0.025 and 0.05 in the simulation-based staircase test. These three values are accordingly investigated with the Normal distribution in cases 1 to 3, since it is the only probability distribution that can be estimated by all evaluation methods, namely DM, MLE and KDE methods. To analyse the effect of the underlying probability distribution, the CV was fixed to 0.025 by the authors in cases 4 and 5. Only the MLE and KDE methods were applied to perform the estimation of the fatigue limit distribution in these cases, since the DM method assumes that the fatigue limit should be a Normal distribution [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF].

The evaluation of the estimate quality is carried out by normalising the estimated median value m k and the estimated standard deviation s k to the true median value m 0 and the true standard deviation s 0 of the given probability distribution, respectively. The 5 th , 50 th (a.k.a. median), and 95 th percentiles were also estimated by the postprocessing method and were used to evaluate the robustness of the compared method performance. In addition, the Jensen-Shannon divergence [START_REF] Nielsen | On the Jensen-Shannon Symmetrization of Distances Relying on Abstract Means[END_REF], denoted as D JS , was used as a measure of difference between the estimated probability distribution and the true probability distribution:

D JS ( P ||Q) = 1 2 D KL ( P ||M ) + 1 2 D KL (Q||M ) with M = P + Q 2 (4.16)
In Eq. 4.16, P denotes the estimated probability distribution from DM, MLE or KDE(Eq. 4.8) method. Q denotes the true distribution initialized in the first step of the simulation (Fig. 4.3). D KL is Kullback-Leibler divergence that defines the relative entropy from Q to P in probability space χ:

D KL ( P ||Q) = x∈χ P (x) log( P (x) Q(x) ) (4.17)

Correction to Scott's rule

Based on this simulation model, we want to find the optimal bandwidth by tuning α in Eq. 4.12. This bandwidth can lead the estimated distribution most approach to the true distribution. The J-S divergence is selected as the factor to evaluate the esti-TEST FOR IMPROVING FATIGUE LIMIT ASSESSMENT mated PDF. The average J-S divergence of 1000 staircase simulated trials with change of α = [1, 1.05, ..., 1.4] are list in Tab. 4.4. The J-S divergence of true distribution and empirical probability density function is selected as the true J-S divergence. In the Tab. 4.4, the first column lists distribution and CV, the second column lists number of specimens and normalised step size. The J-S divergence of KDE with bandwidth approaching to true distribution are listed in third column. The value of α are listed in other columns, and it equals to original Scott's rule when α = 1. The most approaching to the true value are maker with bold text. α = 1.15 is suitable for METHOD most case. Therefore, the bandwidth selector in this study is determined as Eq. 4.18 .

h opt = 1.15 × 4s 5 3n

1 5

(4.18)

Sensitivity analysis of the KDE bandwidth

A sensitivity analysis was carried out in this section to validate the proper bandwidth by comparing different bandwidth selectors. The KDE results with different bandwidth selectors are compared in the following.

Comparison of the median value estimation

It is generally known that the staircase protocol concentrates data near the center [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF], resulting in a high accuracy estimation of the median value of fatigue limit distribution. The comparative study firstly focuses on the median estimation. In Fig. 4.6, the DM, MLE and KDE methods are illustrated by the green, blue and red lines, respectively. The estimation uncertainty decreases with the increase of the specimens number for all three methods. Furthermore, it can be seen from Fig. 4.7 that the step size has little effect on the median value estimation, but a larger step size (normalised d > 1) leads to a small bias of median estimation when n = 30. Since all three methods give a reasonable median estimation, the following study presents only the results for the standard deviation estimation. The divergence of the different statistical methods is mainly reflected in the standard deviation estimation. To prove the robustness of the KDE method, the standard deviation estimates of the number of specimens and step sizes are presented with setup case 1, i.e., Normal distribution and CV=0.025. The results obtained with different numbers of specimens and step sizes are presented in Fig. 4.8 and Fig. 4.9, respectively. The comparison of the three methods suggests the following conclusions:

(1) According to Fig. 4.8, all three methods have relatively stable performance for different numbers of specimens. For each method, the number of specimens has no effect on the 50 th percentile of all estimations but reduces the uncertainty. (2) From the red line in Fig. 4.9, the KDE works better for standard deviation estimations when normalised d = 1 -1.5. (3) According to Fig. 4.9, there are slight differences between DM and MLE methods for a Normal distribution. This observation was also pointed out by [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF]. (4) According to Fig. 4.9(a) and (b), the KDE method shows the closest results to the true value in most cases of step sizes with smaller uncertainty. This makes sense because the correction equation focuses on the small sample size. [START_REF] Zerbst | Applying fracture mechanics to fatigue strength determination -Some basic considerations[END_REF] There is a transition between the normalised step sizes 0.7 -0.8 in Fig. 4.9.

This discontinuity comes from the piecewise correction equation Eq. 4.13. Taking simulation case 1 (n = 30) as an example, the estimated standard deviation (s) and the number of used piecewise equation is shown in Fig. 4.10.

In the upper part of Fig. 4.10, the blue lines represent the results of KDE without correction, and the red lines represent the results of KDE with correction. In the lower part of Fig. 4.10, the red line is the number of first piecewise equation (s > d) used, and the blue line is the number of second piecewise equation (s ⩽ d) used. A total of 1000 trials have been performed in the simulation. The horizon axis is the normalised step size in the range of [0.1-2.0]. It is necessary to note that the green arrow in Fig. 4.10 indicates the crossing point between the blue line and the red line, as well as the discontinuity of 50% estimation. The reason that this point locates between 0.7 and 0.8 instead of 1.0 is that the staircase method has an inherent low biased estimation of the standard deviation [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF][START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF], due to concentrating the majority of the data points near the median. 6) The scatter of the estimation is significantly reduced by applying the KDE method, especially for step sizes in the range of 1.5s 0 -2.0s 0 .

In this case, the KDE method has the slightest uncertainty regarding the number of specimens and the step size. It achieves the goal of reducing the effect of step size and reducing uncertainty.

Comparison of standard deviations with different coefficients of variation (cases 2 and 3)

Since the mean is considered constant, the standard deviation varied with the coefficient of variation CV=0.0125 and CV=0.05. The results of the related standard deviation estimation are plotted against the number of specimens in Fig. 4.11 and Fig. 4.12. It is clear that the KDE method shows the best estimation properties in the case of CV=0.0125 and CV=0.05, which leads to the same conclusions for CV=0.025. With the help of the correction equation (Eq. 4.13), the estimated standard deviation is less affected by CV. The results analyzed with the knowledge of the coefficient of variation highlight the interest of KDE.

Comparison of the standard deviation with

Lognormal and Weibull distributions (cases 4 and 5)

In the previous sections, the KDE method was evaluated with the Normal distribution compared to other evaluation methods. It should be noted that the KDE method requires no prior information about the underlying distribution. Therefore, it is necessary to evaluate the effectiveness of the proposed method with other probability distributions, such as Lognormal and Weibull distribution. Since the DM method is incapable of estimating the non-Normal distribution of the fatigue limit, the KDE method will be compared to the MLE method in this section. The estimation of the standard deviation with respect to the step sizes is shown in Fig. 4.13 for the Lognormal distribution and Fig. 4.14 for the Weibull distribution. From Fig. 4.13, the outcome considering the Lognormal distribution presents the same main conclusions as that of the Normal distribution. From Fig. 4.14, the MLE estimation is not capable of providing reliable information regarding the Weibull staircase data. Focusing the KDE results (red line) in Fig. 4.13(b) and Fig. 4.14(b), the KDE method has better performance for Weibull distribution. From the results of this work, the KDE method combined with the Weibull distribution will achieve better performance for the experimental test. Therefore, the KDE is a distribution-free statistical assessment method that works better than the MLE even if the MLE fits the data distribution perfectly.

Application of KDE method on experimental data

The non-parametric method is applied in this section in order to characterise the fatigue limit of the DC01 low carbon steel specimens by using the data in the up-and-down diagram given in Chapter 2. The estimated PDF and CDF of fatigue limits calculated by the KDE with optimized bandwidth and DM method are shown in Fig. 4.15 and Fig. 4.16. Correspondingly, the median and standard deviation of fatigue limit resulting from the density estimation are shown in Tab. 4.6. (1) The bending fatigue limit of DC01 was obtained in strain domain as 1366.66 µm/m with the standard deviation 12.21 µm/m ; (2) For the present results, three methods use different experimental informations.

The sample size by DM is 18 (failure) while by MLE and KDE are 36 (total). Moreover, the MLE distinguishes the specimens by failure or survival result, while KDE does not require this information. (3) Three methods lead to almost same median estimation. (4) It is apparent that KDE has an estimated results accordance with MLE method.

The median and standard deviation values by KDE and MLE estimations converge to each other. ( 5) All estimated median value are consistent with the observed fatigue strength from ε -N curve in Chapter 2 and comprise in the 95% confidence interval. (6) A larger standard deviation of fatigue limit is obtained from DM. After the correction of the standard deviation in KDE method, the predicted value becomes much smaller.

Conclusion

In this chapter, a non-parametric evaluation method is proposed in this work to estimate the fatigue limit from the staircase test. The method originally combines the Kernel Density Estimation (KDE) with nonlinear bias correction. The performance of the proposed method is optimized by setting dynamically the bandwidth of the KDE. A simulation-based study is carried out to evaluate the estimate quality compared to other widely used evaluation methods like the Dixon-Mood (DM) and Maximun Likelihood Estimation (MLE) methods. In the numerical study, the underlying distribution of the fatigue limit is presumed to be characterized by a couple of probability distributions and coefficients of variation (CV). The numerical experiment involves three probability distributions (i.e., Normal, Lognormal and Weibull) that have been widely studied in related work. The dependence of the estimate quality on the sample size and the step size of the staircase test was investigated. From the numerical results, the proposed KDE-based method outperforms the MLE and DM in terms of estimations of the median, standard deviation and probability distribution of the fatigue limit. For the normally distributed fatigue limit, all the methods investigated (i.e., DM, MLE and KDE) result in a good estimation on the median value. But DM and MLE provide poor estimates of the standard deviation of the fatigue limit, especially for small sample tests, i.e., when the number of specimens is less than 30. In contrast, the proposed method offers an estimate that is relatively closer to the true standard deviation regardless of the number of specimens. This outstanding performance is also observed for the J-S divergence which measures the similarity between two probability distributions. The fatigue limit distribution estimated by the KDE method always returns a smaller J-S divergence, in other words, the estimated distribution is closer to the presumed Normal distribution. Not to mention that the estimate quality of the KDE method is stable with respect to the different CV coupled with the Normal distribution.

The proposed method is also applied to experimental data, which obtain the fatigue distribution from staircase data. With respect to the non-Normal distribution of the fatigue limit (i.e., Lognormal and Weibull), the estimations made by the proposed method are always better than those of the MLE method. The estimated standard deviation of the fatigue limit is really noticeable with the KDE method and the Weibull distribution, nevertheless the estimate quality measured by the J-S divergence is relatively poor. This leads to further research on the kernel functions of the proposed KDE-based method.

In the following chapter, a staircase procedure improvement is presented.

Chapter 5

Bayesian improved staircase experimental design

To address the essential deficiency of the conventional staircase test method, this chapter proposes an optimized test method based on Bayesian Maximum Entropy Sampling (BMES) and Latin Hypercube Sampling (LHS) to determine the fatigue limit distribution of the structure. The proposed method integrates the prior information from samples to obtain the posterior probability as well as the fatigue limit distribution. It reduces the statistical analysis error caused by the small number of samples in the fatigue limit test.

The current chapter is structured as follows: a summary of the staircase problems and several optimizations for test protocol are given in Section 5.1. The background of fatigue limit test, the Bayesian theory, the BMES and the LHS methods are interpreted in Section 5.2, Section 5.3 and Section 5.4, respectively. Then, Section 5.5 introduces the proposed improved Bayes-LHS staircase method based on BMES and LHS. Section 5.6 defines the numerical simulation setup for the conventional staircase method, the Bayesian staircase method and the proposed improved Bayes-LHS method. Lastly, Section 5.7 presents the simulation results, and evaluates the effectiveness of the proposed Bayes-LHS method by comparing the estimated distributions under the same condition. CHAPTER 5. BAYESIAN IMPROVED STAIRCASE EXPERIMENTAL DESIGN

Background

It is well known that the estimated sample median (or mean) from the staircase method is usually much closer to the true value. However, due to the nature of the staircase test, it is challenging to obtain a proper estimation of the sample standard deviation [START_REF] Rabb | Staircase testing -confidence and reliability[END_REF].

The improvement of the accuracy of the fatigue limit estimation can be classified into the evaluation techniques and the experimental protocol. In the last decades, several researchers have investigated ways of post-processing. For example, Pollak et al. [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] formulated a non-linear correction on the standard deviation of fatigue limit and involved bootstrapping sampling for a small number of specimens. Müller et al. [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] compared several evaluation techniques through Monte-Carlo simulation (MCS) and introduced the usage of statistical hypothesis tests in evaluating experimental results. Nevertheless, both methods didn't solve the disadvantage of the dependency to step size of fatigue limit test.

Due to the intrinsic limitations of the staircase method, recent studies focus on improving the test protocol. To date, several studies have focused on experimental design optimization. Wallin [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF] simulated a modified staircase procedure with a small step size to cover all anticipated standard deviation ranges. However, a small step size may lead to larger sample size. Bai et al. [START_REF] Bai | Measurement and estimation of probabilistic fatigue limits using Monte-Carlo simulations[END_REF] proposed a Monte-Carlo method that offers higher test data efficiency and can be variable in step size. However, the step size still needs to be guessed before the test, and the result strongly depends on the step size. Roué et al. [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] developed a new experiment staircase procedure and reduced the uncertainty in the standard deviation estimation by reloading unbroken specimens. Magazzeni et al., Magazzeni et al. [116,[START_REF] Massimo | Bayesian optimized collection strategies for fatigue strength testing[END_REF] proposed protocols using Bayesian Maximum Entropy Sampling (BMES), in which a faster convergence is obtained.

To the moment of writing, the application of parametric models based on robust initial assumptions, enriched by the Bayesian theorem, provides a more reliable fatigue characterization of materials without resorting to black boxes. The Bayesian staircase strategy was firstly proposed by Engler-Pinto et al. [START_REF] Carlos | Statistical Approaches Applied to Fatigue Test Data Analysis[END_REF]. Then, Engler-Pinto et al. [START_REF] Carlos | Statistical Approaches Applied to Fatigue Test Data Analysis[END_REF] incorporated the Bayesian staircase strategy into Life-Regression Models (S-N curve) [START_REF] Jr | Statistical approaches applied to very high cycles fatigue[END_REF]. Alcalá-Quintana et al. [START_REF] Alcalá | A comparison of fixed-stepsize and Bayesian staircases for sensory threshold estimation[END_REF] carried out a numerical study to compare the fixed-step-size and the Bayesian staircase method. The authors found that the standard deviation of Bayesian estimates is systematically lower than that of the conventional staircase in the same condition.

Based on the idea of Bayesian theory in the experimental test [START_REF] Li | A Bayesian Optimal Design for Sequential Accelerated Degradation Testing[END_REF][START_REF] Babuska | Bayesian inference and model comparison for metallic fatigue data[END_REF], an application is carried out by using maximum entropy sampling as a criterion for choosing experiments to maximize the gain in information regarding prediction at unsampled 5.2. BAYESIAN THEOREM IN FATIGUE TEST sites. Sebastiani et al. [START_REF] Sebastiani | Maximum entropy sampling and optimal Bayesian experimental design[END_REF] introduced an extension of the Bayesian principle to the estimation problems. Magazzeni et al. [START_REF] Massimo | Bayesian optimized collection strategies for fatigue strength testing[END_REF] described the improved staircase method using Bayesian Maximum Entropy Sampling (BMES) with details.

According to these studies, the Bayesian staircase method requires very little information about the sample set prior to the testing. The Bayes and the LHS are proposed to improve the conventional staircase tests in this work. The LHS [START_REF] Karolczuk | Fatigue life uncertainty prediction using the Monte Carlo and Latin hypercube sampling techniques under uniaxial and multiaxial cyclic loading[END_REF] are incorporated into BMES to make sampling more uniform in the cumulative density probability domain. The current chapter focuses majorly on the implementation of BMES and LHS for improving conventional staircase protocol, in which the stress step size is not constant and is calculated after every test.

Bayesian theorem in fatigue test

Let be a random data set x = {x 1 , x 2 , ..., x n } that denotes the specimens from fatigue limit test, x is subject to a PDF P (x|θ) where, θ is model parameter, and x is the fatigue limit (test results in fatigue test). Bayes rule is built as:

P (θ|x) = P (x|θ)P (θ) P (x) (5.1)
where, P (x|θ) is the likelihood function, P (θ) is the prior probability, P (x) is the observations from experimental test (a.k.a, marginal probability), P (θ|x) is the posterior probability on the left side of the equation.

Model parameter

In the practical computation, the matrix θ includes all possible model parameters that are described by shape, location, and scale (see Appendix E). For example, the θ in Normal distribution is created as:

θ =       (m 1 , s 1 ), (m 2 , s 1 ), . . . , (m a , s 1 ) (m 1 , s 2 ), (m 2 , s 2 ), . . . , (m a , s 2 ) . . . , . . . , . . . , . . . (m 1 , s b ), (m 2 , s b ), . . . , (m a , s b )       (5.2)
where, m and s are the location and scale for Normal distribution . The mean range and standard deviation range are discredited with the number of a and b. For example, a = 100 and b = 100 are used in the following analysis.

The model parameters also define all possible stresses for tested specimens. That is,

S a = [m 1 , m 2 , . . . , m max ].
Each stress amplitude S a corresponds to a result x (failure or survival).

Likelihood with censored data

Considering the one result x from a staircase test, the likelihood that contains information in test results (e.g., failure or survival) is:

L = P (x|θ) = Sa,up S a,low f (S a , θ)dS a = F (S a , θ) (5.3)
where, f (θ) and F (θ) are the PDF and CDF for the model parameter θ, S a is the stress amplitude, S a,low and S a,up are the boundary limits of the stress S a . For example, based on the Normal distribution, the model parameter θ consists of all possible median and standard deviation values. The aim is to calculate the CDF related to a chosen stress amplitude S a based on the model parameters [m, s] in θ. In case of Normal distribution, F is a CDF that has same dimensions with θ.

F = Φ(S a , θ) (5.4)
In practice, it should be distinguished between the survival stress amplitude and the failed stress amplitude and calculate separately the CDF with these stress amplitudes. There are two kinds of likelihood on one specimen with left-or right-censoring:

L i = L f i = F (S a , i) L s i = 1 -F (S a , i) (5.5)
where, i is the ordinal number of the specimen, S a , i is the stress amplitude of i th specimen.

Total likelihood with all previous specimens x = x imax , x 2 , . . . , x imax :

L = P (x|θ) = imax i=1 L i (5.6)
In computational practice, the likelihood (L) is in the logarithm form. With the prior in logarithm, the Eq. 5.6 is carried out as follows:

ln L = ln P (x|θ) = imax i=1 ln L i (5.7)

Prior and posterior

The prior and posterior are usually done in the logarithm form to avoid numerical issues with small values. The prior (or posterior) is built as a discrete matrix with the same dimensions as θ.

BAYESIAN THEOREM IN FATIGUE TEST

Here we present how to update the posterior by prior after a single test. During the staircase test, after i th specimen has been tested, the prior before i th specimen becomes:

P (θ) = P (θ|x 1:i-1 ) (5.8)
In the logarithmic form:

ln P (θ|x 1:i-1 ) = t=i-1 t=1 ln P (θ|x t ) (5.9)
where t is the iteration number.

In the experimental staircase test, the x and θ are discrete values, and the total probability equation is:

P (x) = θ P (θ) • P (x|θ) = θ P (θ|x 1:i-1 )P (x i |θ)
(5.10) Taking Eq. 5.8 and Eq. 5.10 into Eq. 5.1, the update of the prior for the i th specimen P (θ|x 1:i ) according to P (θ|x 1:i-1 ) is described as:

P (θ|x) = P (x|θ)P (θ) P (x)
P (θ|x 1:i ) = P (x i |θ)P (θ|x 1:i-1 )

θ P (θ|x 1:i-1 )P (x i |θ)

(5.11

)
where i is the specimen ordinal number. The posterior P (θ|x 1:i ) calculated from the prior of the previous specimens P (θ|x 1:i-1 ). P (x i |θ) is the likelihood L i . In practice, incorporating Eq. 5.7 and Eq. 5.9, the update of posterior (Eq. 5.11) in logarithm is described as:

ln P (θ|x 1:i ) = ln[P (θ|x 1:i-1 )P (x i |θ)] - θ ln[P (θ|x 1:i-1 )P (x i |θ)]
(5.12)

In order to simplify the description, a normalized matrix M is defined by:

M = ln[P (θ|x 1:i-1 )P (x i |θ)] = ln[P (θ|x 1:i-1 )] + ln[P (x i |θ)] (5.13)
It must be ensured that the element of M is greater than 0. For the element less than 0, it is set to a tiny number (such as 1 × 10 -7 ).

CHAPTER 5. BAYESIAN IMPROVED STAIRCASE EXPERIMENTAL DESIGN

With Eq. 5.13, Eq. 5.12 can be written as:

ln P (θ|x 1:i ) = M - θ M (5.14)
The posterior after i th specimen is the prior for (i + 1) th specimen.

It is necessary to define a initial prior matrix before the test. The fully flat initialised to the prior has been proved to work better in convergence [START_REF] Magazzeni | Bayesian Optimised Collection Strategies for Fatigue Testing -Constant Life Testing[END_REF]. A flat prior is equivalent to assuming no prior at all, which means it is created by an array of equal values. Consistent with the calculation of posterior in Eq. 5.11 and Eq. 5.14, the matrix M , as described in Eq. 5.13, is initialised as:

M = J a,b (5.15)
where a, b are the dimensions of θ, the J a,b is all-ones matrix of the same dimension as θ. The initial prior is obtained by taking Eq. 5.15 into Eq. 5.14.

Bayesian maximum entropy sampling

Bayesian Maximum Entropy Sampling (BMES) contributes to predicting the expected result by maximizing the entropy gain. After testing i specimens, the stress amplitude for (i + 1) th specimen can be selected by maximizing the expected gain in Shannon information in the posterior.

Shannon entropy

The Shannon entropy measures the uncertainty of the system. Shannon information I included in the posterior is:

I(P (θ|x)) = -ln(P (θ|x)) (5.16)
This information indicates the uncertainty of the posterior P (θ|x). According to this information, the Shannon entropy is defined as:

H(P (θ|x)) = E(I(P (θ|x))) = - θmax θ min dθP (θ|x) ln P (θ|x) (5.17)
where we see that the Shannon entropy is the product of the probability and the uncertainty. When taken from a discrete θ and finite samples x, the Shannon entropy 102 5.3. BAYESIAN MAXIMUM ENTROPY SAMPLING formula can be explicitly written as follows:

H(P (θ|x)) = - θ P (θ|x) ln P (θ|x) (5.18)
After the i th specimen, we need to find the proper stress amplitude for the next (i + 1) th specimen based on Bayesian progression.

In order to obtain an optimized stress amplitude, the objective is to maximize the information of the test stress amplitude. The entropy for (i + 1) th specimen is defined as:

H(P (θ|x 1:i+1 )) = - θ P (θ|x 1:i+1 ) ln P (θ|x 1:i+1 ) (5.19)
where P (θ|x 1:i+1 ) is posterior for the (i+1) th specimen with design of stress amplitude S a and result x i . The model parameters θ are kept unchanged during the staircase test.

Note: The gain in entropy (∆H, entropy difference between the (i + 1) th and i th specimen) can also be used to replace the Eq. 5.19. However, the same results are obtained but with more computation.

Shannon entropy provides the information of staircase data but not the quality of the estimation results. The minimum Shannon entropy may not consist of the best posterior for estimation. The minimum Shannon entropy could not be used as a stopping criterion for the test.

Utility function

After testing i specimens, the next stress is chosen to maximize the expected information gained by sampling at a subsequent specimen x i+1 .

In the staircase test, a utility function is built by considering the expected information with both results of a specimen, survival or failure, at the expected stress amplitude (a stress amplitude possible to be tested). The utility function with the stress amplitude S a,i+1 is defined as:

U (S i+1 ) = H s (S a,i+1 , x)p s (x|S a ) + H f (S a,i+1 , x)p f (x|S a ) (5.20)
where, U is the utility function that describes the entropy gain with the result x i+1 .

x is a vector of all previous tests. U , H, p are all (float) values. The superscript "s" represents the "survival" and superscript "f " represents the "failure".

Similar to Eq. 5.18, the Shannon entropy is expressed as Eq. 5.21:

H s (S a,i+1 , x) =θ P (θ|x) ln P (θ|x) with survival H f (S a,i+1 , x) =θ P (θ|x) ln P (θ|x) with failure (5.21) The difference between these two equations is posterior P (θ|x), which is calculated with Eq. 5.11. Two cases, including survival and failure, are considered for posterior with different likelihood as Eq. 5.5.

It should be noted that the probabilities p s and p f are the multiplication of the likelihood and the prior, after integrating to reach the marginal probability.

p s (S a,i+1 , x 1:i ) = θ [P (θ|x 1:i+1 ) • (1 -Φ(S a,i+1 ))] p f (S a,i+1 , x 1:i ) = θ [P (θ|x 1:i+1 ) • Φ(S a,i+1 )] (5.22)
To summarise, the calculation of the utility function is illustrated in the flowchart of Fig. 5.1. In this study, the utility function is calculated by the differential evolution optimization method, and the stress corresponding to maximum utility is the expected load level for the next specimen.

BAYESIAN MAXIMUM ENTROPY SAMPLING

Bayesian staircase method

By applying the utility function to select the stress amplitude, the Bayesian staircase procedure [START_REF] Magazzeni | Bayesian Optimised Collection Strategies for Fatigue Testing -Constant Life Testing[END_REF] can be shown as Fig. 5

i P x - θ 1: 1 ( | ) i P x - θ 1: 2 ( | ) i P x  - 1:2 ( | ) P x  1 : 1 Test Posterior 1: 1 ( | )
( | ) Assuming that the number of specimens is n = 30, an example of an up-and-down result is shown in Fig. 5.3. It can be seen that the first test is searched as the center of the mean range because there is no prior information. The second test is located closer to the boundary in a large load amplitude. Then, the load amplitude gradually decreases, until the opposite result appears. After that, the tests converged to near the mean value. All load amplitudes are optimized by the utility function as described before. So we can find variable load amplitudes during the simulated staircase test. In this study, the number of specimens n = 30, 35, 40, 50, 70, 100 are simulated, and similar results are found. The variable step size is used to maximize the utility function that based on Shannon entropy and probability.

From the Fig. 5.3, the Bayesian staircase method has two shortcomings. Firstly, this method relies on BMES to select the stress amplitude but ignores the test results (failure or survival). Secondly, the stress amplitudes gradually converge to the same values. Similarly, as shown in Fig. 5.4, Shannon entropy has a slower decrease rate after several tests. The fatigue tests in the later stages provide less information for searching for the best fit distribution. In order to improve the Bayesian staircase

Improvement by using Latin Hypercube Sampling

Latin Hypercube Sampling (LHS) is a statistical method for generating a nearrandom sample of parameter values from a distribution [START_REF] Michael | The generalization of Latin hypercube sampling[END_REF]. As one-dimension LHS sampling strategy, the cumulative distribution function [0, 1] is firstly divided into N equal parts to get [0, 1/N ], [1/N, 2/N ], ..., [(N -1)/N, N ]. For these N partitions, a sampling point can be randomly selected in each partition, and then obtains N sampling values. This strategy maintains the independence between the samples at the same time.

The LHS has good uniformity with regard to the individual sub-layer. The CDF and PDF of an example of LHS are presented in Fig. 5.6. The 10 points sampled from a Normal distribution subject to N (400, 10 2 ) in CDF and PDF can be shown in Fig. 5.6.

In the Fig. 5.6(a), the CDF curve is divided into 10 non-overlapping layers along y-axis (the blue lines). So that each interval (red lines) has the same cumulative probability. Selecting centring points (green points) in each interval, and total 10 sampling results (purple points) are obtained. Correspondingly, these sampling data with PDF is shown in the In protocol design, the sampling procedure takes into account the previous stress amplitude. The LHS divides the CDF into N intervals and samples N points. To maintain uniformity during all the tests, the interval with the least number of specimens is selected as the stress amplitude.

An assumed distribution is needed to use the LHS in the experimental design. Hence, the LHS is used after BMES. Besides, the LHS requires less calculation and works faster than BMES 5.5. Application of the Bayes-LHS method for staircase representation

Input parameters

For the Bayes-LHS staircase test protocol, the parameters that need to be set before the test are:

(1) The distribution type. In this work, the Normal distribution is chosen to conduct the simulation study, though other models such as Weibull distribution and Lognormal distribution are also examined. (2) The possible range of the median of the fatigue limit distribution. This proposed method is not sensitive to predefined ranges. In the practical application of this protocol, the median range in model parameter can be selected from a small value to yield strength. where m is the median and s is the standard deviation. Based on the input range, all possible model parameters (θ) for the fatigue limit distribution are created as a combination of the shape, location and scale according to the specific distribution initially assigned.

Bayes-LHS staircase protocol

The flowchart of the Bayesian staircase procedure is presented in In Fig. 5.7. The specimens tested by BMES are indicated as light blue, while the LHS are indicated as light red. i is the ordinal number of the specimens. The improved experimental procedure includes the stress amplitude determined from BMES and LHS. ( | ) For the first step (first specimen), the initial prior is created as described in Section 5.2.3. The possible stress values are chosen from the median range of the input parameter. The utility function, as described in Section 5.3.2, is used for searching this stress amplitude. After experimental test, the stress amplitude and test result (survival or failure) are obtained, and the posterior P (θ|x 1:i+1 ) is updated based on this test result. This posterior is the prior for the next specimen.

Numerical simulation of the Bayes-LHS test protocol

In this study, the possibly unknown distribution is asymptotically approached by the Normal distribution. However, other distribution models, such as Lognormal distribution and Weibull distribution, can also be used as the underlying distribution.

In order to evaluate the effectiveness of the proposed method, a numerical simulation modelling the staircase test is carried out. The staircase simulation is configured as shown in Fig. 5.11.

Initial parameter

Median m0

Standard deviation s0

Weibull

Staircase simulation

Fatigue limit estimation

Results and discussions

A total of 400 trials are conducted for the Bayesian staircase and the Bayes-LHS staircase, respectively. Firstly, the evolution in the Shannon entropy indicates the efficiency of the test. Taking the n = 30 as an example, the decrease of Shannon entropy within a test from 400 simulations is shown in Fig. 5.12. In the Fig. 5.12, the blue area represents 5 th -95 th percentile of the Shannon entropy from the Bayesian staircase method, while the red area represents that from the Bayes-LHS method. The blue and red lines are 50 th percentile of all entropy values. It can be found that both methods give the same tendency for entropy to decrease. The entropy value decreases fastest for the first several specimens, which also have the same results because the both methods start from the same model parameters (θ).

Sensitivity analysis of input parameters

To reach the better solution, it is necessary to analyse their sensitivity to input parameters, including guessed median range and standard deviation range. This study chooses three different median ranges and three standard deviation ranges for the simulation work, as listed in Tab. 5.1. Note that the true distribution is assumed as a Normal distribution [400, 10 2 ]. 200 trials are simulated for the comparison. The estimated results for the median and the standard deviation are compared in Fig. 5.13 and Fig. 5.14 which show the effect of median range and the effect of standard deviation range, respectively. From Fig. 5.13, the difference occurs for a number of specimens less than 20. For the Bayes-LHS test with a number of specimens greater than 20, the input median range has almost no influence on the estimation results. It is possible to choose between a small value and the yield stress as the median range in a real experiment. The same findings can be observed in Fig. 5.14. The estimated standard deviation does not rely on the guessed standard deviation range.

In a word, the Bayes-LHS method is not sensitive to input parameters with a sufficient number of specimens in the test (n ⩾ 20).

Comparison of the Bayesian staircase and Bayes-LHS method

Choosing Case 4 of Tab. 5.1 as input, the estimated results for median and standard deviation is shown in Fig. 5.15. In Fig. 5.15, the Bayesian staircase and Bayes-LHS method are illustrated by the blue line and the red line, respectively. It can be observed that:

(1) Similar to the conventional staircase method, the estimated results of the Bayes staircase and Bayes-LHS method become more accurate with more specimens. (2) From Fig. 5.15(a), the Bayes-LHS method is more stable on the median estimation. The estimated median is centred on the true value. (3) From Fig. 5.15(b), the estimated standard deviation obtained from the Bayesian staircase method and Bayes-LHS method have almost the same uncertainty. More-5.7. RESULTS AND DISCUSSIONS over, both methods have an underestimated bias for the standard deviation, especially with fewer number of specimens.

Comparison to the conventional staircase using KDE

In order to show the advantage of the Bayesian staircase method, the comparison between Bayes-LHS method and the conventional staircase method using KDE is presented in this section. In this comparison, the number of specimens is fix at 30. For the step size only existed in the conventional staircase method, the normalized step size 0.5, 1.0, 1.5 are selected for comparison. 200 trials are conducted for each method.

For Bayes-LHS method, the input parameter of Case 4 (see Tab. 5.1) is adopted. The conventional staircase is simulated as the procedure given in Section 4.2. For comparison with respect to the number of specimens n = 30 (Fig. 5.16), we choose the normalised step size d=0.5, 1.0 and 1.5 as examples. Since there is no step size in the Bayes-LHS method, the results are presented for comparison with respect to the step size (Fig. 5.16(b)). Based on 200 trials, the estimated median and standard deviation is shown in Fig. 5.16 and Fig. 5.17 (1) From Fig. 5.16(a), four line kinds almost overlap. The estimated median results obtained from the Bayes-LHS method and KDE are almost identical. The Bayes-LHS method is worse when the number of specimens is less than 20. The same observation can also be made for Fig. 5.17 includes the true value, while that of the conventional staircase is affected by the step size. However, the vertical red line is longer than others, which means that the estimation uncertainty of Bayes-LHS method is lower than KDE with correction. (3) From Fig. 5.17(b), the results of the conventional staircase method are dependent on the step size even though the correction is applied to KDE. (4) From Fig. 5.16(b) and Fig. 5.17(b), the horizontal red dash are below the horizontal black line. This means, the Bayes-LHS method has a disadvantage of underestimating the standard deviation, especially when the number of specimens is in the range of .

Conclusion

This chapter presents an improved experimental staircase procedure, Bayes-LHS method, based on Bayesian Maximum Entropy Sampling (BMES) and Latin Hypercube Sampling (LHS). This method uses the BMES and LHS to select the stress amplitude (load level) in staircase testing, which removes the step size parameter. A simulation is conducted to compare the Shannon entropy and estimated median and standard deviation for the numerical validation of this new protocol.

Compared to the conventional staircase method, the advantage of Bayes-LHS method can be listed as follows:

(1) The Bayes-LHS method avoids the use of guessed start stress and fixed step size. It should artificially select a range of possible model parameters before the Bayesian staircase test. (2) Due to the non-sensitivity of input parameters, very little information is required on the sample set before the test. (3) The conventional staircase method with corrected KDE has an advantage in re-ducing the estimation uncertainty. Therefore, it is suitable if the standard deviation is well-guessed. Otherwise, the Bayes-LHS method is recommended for the fatigue limit distribution assessment. (4) There is no need to use a post-processing method. The distribution model parameter is obtained directly from the posterior.

Chapter 6

Conclusion and perspective work

The present work depicts the subject of fatigue limit assessment. The thesis covers aspects from mechanical fatigue experiments to statistical methods. This final chapter summarises what we have learned from previous works and points out the most promising directions for future research.

According to the literature review, fatigue limit assessment is a challenging practical problem as the structure is exposed to vibration conditions, thus is crucial a reliable structural design. To achieve probability distribution of the fatigue limit, the critical challenges come from the experimental test method and an advanced evaluation technique of the test data.

The fatigue limit assessment begins with the experiment test. A vibration bench is a reasonable solution to reduce fatigue test time. In contrast, the fatigue limit test requires the specimen to be subjected to a constant stress amplitude for a certain number of cycles. A strain control technique applied to an electro-dynamic shaker is proposed to conduct a constant strain amplitude test. The efficiency of the proposed approach has been demonstrated by testing in bending DC01 steel plates at their first resonant frequency. The strain control on resonance (amplification of the signal) and relatively high-frequency excitation leads to a reduction of the testing time to reach a large numbers of cycles. This approach is therefore effective for testing the staircase method. The fatigue threshold and the scatter of steel DC01 is estimated by the staircase results.

The statistical method offers approaches to provide the fatigue limit distributions for each staircase test. The key question is: what results from a re-conducting execution of the staircase test? Uncertainty analysis is carried out to evaluate the fatigue limit distributions of a material obtained from the staircase test. For this purpose we applied the resampling method, leave-one-out and bootstrap on the staircase data in order to deduce the scatter of the mean and standard deviation of the distribution. The results of this study highlighted the high inherent uncertainty in the standard deviation estimation.

To reduce this uncertainty, we propose to use the Kernel density estimation (KDE) in this study due to its non-parametric and independence from the distribution model. To compare it with other assumption-based methods, the KDE is tested on different distributions to validate its efficiency. The dependency of the KDE hyperparameter is also studied and optimized to improve its performance in in assessing fatigue limit. Moreover, as the staircase method requires to define an initial step size, a non-linear corrected factor is formulated to reduce its influence in estimating the standard deviation. The numerical approach uses the Monte-Carlo simulation and allows to examine the effect of the number of specimens and the step size. The estimation performance is evaluated on the mean and standard deviation of the fatigue limit involving different distributions and coefficient of variation.

Intending to solve the limitation of step size in the staircase method, a Bayes-LHS staircase protocol is presented to remove the step size in the conventional staircase. The Bayesian theory in fatigue test and Bayesian Maximum Entropy Sampling (BMES) are detailed, and then the Bayesian staircase protocol is provided with an example. This study called Bayes-LHS method enriched the Bayes approach with the Latin Hypercube Sampling (LHS). This proposal is described for staircase procedure. The results are sufficiently optimistic to consider that the Bayes-LHS protocol could replace the conventional staircase method for fatigue limit estimation.

The original contributions of our work mainly involve:

1. For the present approach, staircase tests for deducing the fatigue limit are accessible by using the strain control method. The strain control is effective for low-carbon steel and provides a stable vibration control method to reach the High cycle fatigue domain in a relatively short time. 2. The bootstrap resampling is applied to evaluate the uncertainty of the staircase test. It provides a numerical sampling method to avoid the real experimental tests. 3. A non-parametric evaluation method based on Kernel Density Estimation (KDE) estimates the fatigue limit distribution in a data-driven way rather than using handcrafted heuristic strategies. The proposed method estimates the fatigue limit distribution without prior knowledge. It is not sensitive to its hyperparameter, and is less affected by the test parameters of the staircase method, such as the number of specimens and the step size. 4. Bayesian including the LHS for optimizing the staircase method to eliminate the effect of the step size. The improved test procedure uses the previous information during the test, and requires less information before numerical testing. Also, there is no need to use a post-processing method such as Maximun Likelihood Estimation (MLE) or KDE. Because the distribution model parameter is obtained directly from the posterior.

In the experiment part of this study, low carbon steel specimens were selected to carry out fatigue tests with strain control. We found that this material is the simplest to study with this type of excitation (control of the deformation in vibration). We had previously tested Ti-6Al-4V and steel 304L. However, for some unknown reasons (different hypotheses were put forward but none of them gave satisfaction), it was possible to test materials in acceleration but not in strain control. In fact, the balance point (zero) of the strain measurement is shifted during the test.

Lots of Ti-6Al-4V and steel 304L specimens were tested with strain control. Some Ti-6Al-4V specimens with different treatments are presented as examples in Fig. 6.1. In this figure, different treatment for specimens and load strain amplitude are list in Tab. 6.1 From the Fig. 6.1, the obtained strain signals show a decrease with the number of cycles. The higher strain amplitude for the specimens with the same treatment leads to a faster decrease. Comparing C1 and other cases, the vibration stress relief and the heat-treatment before the track dwell test did not improve the test performance.

Indeed, we have found that strain control is not possible on Ti-6Al-4V titanium alloy and 304L stainless steel. One of the reasons is stated by Barbier [124] who indicates that 304L has a cyclic behavior that evolves over several cycles and does not stabilize even beyond several million cycles. Therefore, we preferred to direct this study toward standard steel. Hence, a research is required to evaluate the applicability of strain control for different materials.

To improve the study, several other ways will be investigated for further work:

1. Another investigation concerns the effect of frequency and geometric dimensions of the specimens on the fatigue limit. Because this research could lead to a "perfect" specimen to reach the fatigue limit distribution 2. An improvement of the Bayes-LHS approach is necessary to achieve a wellperforming approach. For example, a way is to use the chaos polynomial to fit the distribution. The interest can be to adjust the coefficients on the basis of the uncertainties related to the experimental tests (e.g. the clamp). 211.

A.2.4. Location of strain gauge

There is an important experimental particularity about strain gauge which must be addressed. For the first mode, the strain is homogeneous as shown in Firstly, the stress at two different locations can be calculated from a harmonic analysis. The maximum stress at the notch edge is about 1.325 (tension) and 1.06 (bending) higher than the stress in the middle of the specimen, as can be seen in Secondly, Ansys results were also verified by testing. Two strain gauges are mounted in the center and on the edge of the reduced section of one specimen. The sine sweep test and SRTD are performed respectively to compare responses from these two locations of interest. The strain responses in the sine sweep test with acceleration control where s SV represents the Svensson-Lorén corrected standard deviation estimate, and n is the total number of specimens. The equation can improve the low bias of the estimated standard deviation. However, it only considers the sample size and only increases the standard deviation estimated from DM method.

%RF C = 1 %RF C = 2 %RF C = 3 %RF C = 4 %RF C = 5 (-) N end (×10

Pollak's Correction (PO)

Based on the Svensson-Lorén equation, Pollak [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF] conducted lots of simulation works and proposed a non-linear form including step size.

s P O = As DM ( n n -3 )(1.2 s DM d ) b (C.6)
Where, the parameters A and b are constant dependent on the number of specimens (as shown in Tab. C.1).

Braam-Zwaag Correction (BZ)

Braam and Zwaag [START_REF] Jj Braam | A Statistical Evaluation of the Staircase and the ArcSinP Methods for Determining the Fatigue Limit[END_REF] proposed a correction equation that accounts for both the sample size as well as the step size to address this standard deviation bias. 1) The uncertainty of mean and standard deviation will decrease with the increase in the number of samples; 2) the estimation of standard deviation is highly affected by the step size. it is not considered for following comparative study.

s BZ = s DM + d × ( s DM n -0.

D.1. MLE METHOD

where 'ln' holds for natural logarithm in this study. The gradient descent methods, such as Newton method, are frequently used to find the extreme value. This study used Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to find the optimum global resolution of the likelihood function. For the staircase data, it should be noted that the MLE with censoring is only suitable for load levels greater or equal to 4 (j ⩾ 4).

Corrections to MLE

Müller [START_REF] Müller | Accuracy of fatigue limits estimated by the staircase method using different evaluation techniques[END_REF] has concluded that the MLE has a better estimation quality, and he also gave a coefficient(Eq. D.6) to correct logarithmic standard deviation.

s M U L = n -1 n -6.5 × s M LE (D.6)
This correction is only for logarithmic standard deviation, so it is not considered in this study.

D.2. Comparison results and discussions

Using the simulation described in Section 4.3.1, the median and standard deviation estimation is compared in this section. Firstly, the simulation based on Normal distribution, Lognormal and Weibull distribution will be discussed later. The median value from MLE method always approach to the true value and almost independent to the number of specimens and the step size. Next, we will focus on the standard deviation estimation and its corrections. From the results of Normal distribution, some conclusions can been drawn:

1) The uncertainty of median and standard deviation will decrease with the increase of number of samples; 2) Both classical MLE and the censoring MLE give perfect estimation to median value. and it is independent to the number of samples and step size. 3) For the classical MLE method, the estimation of standard deviation is highly affected by step size.

D.3. Conclusion

There is no doubt that the MLE with censoring data leads to a better results than classical MLE. Though the correction is not perfect in all case, but it is suitable for most estimation problems.

E.2. Lognormal distribution

The failure rate curves of Lognormal distribution regularly increase with the fatigue cycling with inherently positively skewed PDF.

The PDF and CDF of Lognormal distribution are given as ref:

f (x) = 1 xλ √ 2π exp    - ln x β 2 2λ 2    F (x) = Φ ln(x) λ (E.3)
where λ is the shape parameter (and is the standard deviation s of the Lognormal distribution), β is the scale parameter (and is also the median of the population).

If we want to create a Lognormal distribution from desired population with mean µ X and s X :

µ = ln µ 2 X µ 2 X + s 2 X s = ln s 2 X µ 2 X λ = s β = exp (µ) (= m X ) (E.4)
The statistical parameters of Lognormal distribution can be obtained:

µ X = exp µ + s 2 2 s X = [exp (s 2 ) -1] × µ 2 X m X = exp (µ) (E.5)
As shown in Fig. E.2, The Lognormal distribution with desired mean 400 and desired standard deviation 10 are created with 100000 data.

For the observed data, µ X = 400.0116 and s X = 10.0249, m X = 399.9219.

E.3. Weibull distribution

The PDF and CDF of two-parameter Weibull distribution is given as:

Figure E.2: Lognormal distribution test f (x) = γ β x β γ-1 exp - x β λ F (x) = 1 -exp - x β λ (E.6)
The statistical parameters of Weibull distribution can be obtained:

µ X = βΓ(1 + 1 λ ) s X = β Γ(1 + 2 λ ) -[Γ(1 + 1 λ )] 2 m X = β(ln 2) 1 λ (E.7)
The shape and scale parameters of the Weibull distribution can be calculated using desired population with mean µ X and s X :

λ : s 2 X µ 2 X - Γ(1 + 2 λ ) (Γ(1 + 1 λ )) 2 + 1 = 0 β = µ X Γ(1 + 1 λ ) (E.8)
In the fatigue analysis, the shape parameter (λ) of a Weibull distribution should be greater than 1 in order to have the meaning of failure rate curve increases with the fatigue cycling. Generally, for λ < 2.6 the Weibull PDF is positively skewed (has a right tail), for 2.6 < λ < 3.7 its coefficient of skewness approaches zero (no tail). Presque tous les équipements mécaniques sont soumis à des vibrations. Les charges vibratoires transférées aux structures réduisent la durée de vie et entraînent des fissures dues à la fatigue. Par conséquent, l'exigence d'une distribution probabiliste de la fatigue est critique en ce qui concerne la sécurité des structure. Comme la limite de fatigue indique l'amplitude de charge endurable sous un nombre de cycles suffisamment élevé, l'obtention de la limite de fatigue d'un matériau est cruciale pour la conception de la fatigue structurelle. Cependant, la machine expérimentale, le protocole d'essai et la méthode statistique posent certains problèmes.

La limite de fatigue est un paramètre important qui reflète la performance des matériaux en matière de fatigue. L'évaluation de la limite de fatigue dans la conception technique est encore fortement basée sur le test expérimental, en particulier la méthode de l'escalier. Cependant, l'essai conventionnel présente des limites intrinsèques concernant la dérivation des données et son influence sur la caractérisation de la fatigue. Pour surmonter ces limites dans l'évaluation de la limite de fatigue, les solutions appropriées pourraient résider dans la machine de test, le post-traitement des données et la stratégie d'essai. Les objectifs sont de réaliser l'essai de fatigue sur agitateur électro-dynamique et de réduire l'erreur d'estimation de la limite de fatigue dans l'évaluation des données et le protocole d'essai.

Cette thèse développe des expériences et des statistiques basées sur la méthode de l'escalier pour l'évaluation de la distribution des limites de fatigue. Elle détaille le processus de conception comprenant le but du développement de contrôle de la déformation pour les tests de fatigue, l'évaluation de l'incertitude, une méthode statistique initiale pour le post-traitement et l'optimisation de la conception du protocole de test en utilisant l'algorithme de Bayes.

Les principaux axes de recherche de ce document sont organisés comme suit :

Chapter F. Cette thèse se conclut avec des perspectives de travail (Chapter F.7).

F.2. Chapitre 1 : Vue d'ensemble de la limite de fatigue

Lors de la sélection d'un matériau pour la conception, il est crucial de connaître les conditions de fonctionnement auxquelles il sera soumis. Dans des conditions de chargement cyclique, la distribution statistique de la limite de fatigue est généralement requise pour délimiter la plage de fonctionnement en cas de fatigue à un nombre de cycle élevé. La limite de fatigue est un niveau de contrainte en dessous duquel la rupture par fatigue ne se produira pas même sous un nombre suffisamment élevé de cycles de charge appliqués à la structure.

Depuis longtemps, la limite de fatigue est considérée comme une propriété essentielle du matériau. C'est une constante matérielle utilisée pour évaluer la valeur d'un matériau et un paramètre nécessaire à la conception de structures résistantes à la fatigue. Dans la conception mécanique d'un matériau, il est important de comprendre F.2. CHAPITRE 1 : VUE D'ENSEMBLE DE LA LIMITE DE FATIGUE non seulement la limite de fatigue moyenne, mais aussi la variance de cette résistance et la probabilité de rupture pour une certaine amplitude de contrainte spécifique.

En ingénierie, la courbe S-N et la limite de fatigue sont souvent utilisées pour évaluer la performance des matériaux et des structures en termes de fatigue. Les spécimens sont testés à différents niveaux de contrainte et le nombre de cycles correspondant (durée de vie en fatigue) est mesuré. Les résultats expérimentaux sont habituellement présentés sous la forme d'un diagramme logS -logN (voir Fig. , il est normal que la limite de fatigue soit déduite de la courbe S-N. Cette dernière peut obtenir une valeurde limite de fatigue mais ne peut pas fournir une distribution statistique. En considérant la limite de fatigue comme une valeur aléatoire, la méthode de l'escalier [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] est beaucoup plus adaptée à la précision de l'estimation de la limite de fatigue.

Dixon et Mood [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] ont proposé une méthode classique de test de sensibilité en 1948 : méthode de l'escalier. Elle est largement utilisée dans les secteurs de la machinerie, de la médecine, de la pyrotechnie, etc. Avec cette méthode, les essais de fatigue sont effectués séquentiellement sur des spécimens soumis à un cycle de contrainte d'amplitude constante jusqu'à un nombre prédéterminé de cycles N L . La première éprouvette est testée à une amplitude de charge initiale choisie arbitrairement. En supposant que l'éprouvette "survive" (runout) jusqu'à N L cycles, l'amplitude de charge appliquée à l'éprouvette suivante est augmentée d'un pas d. Inversement, si l'éprouvette est marquée comme "défaillante", alors l'amplitude de la charge appliquée à l'éprouvette suivante sera diminuée d'un pas d. La taille du pas est généralement constante pendant toute la durée de l'expérience. Cette procédure est répétée en séquence. Les niveaux de charge étant augmentés et diminués par paliers jusqu'à ce que le nombre de spécimens soit atteint. Par conséquent, l'approche de l'escalier fournit une estimation raisonnable de la limite de fatigue médiane, car environ la moitié des spécimens échouent et les autres non [START_REF] Dixon | Staircase bioassay the up-and-down method[END_REF].

En résumé, la méthode d'essai en escalier a été largement utilisée pour évaluer la distribution des limites de fatigue. Elle présente deux caractéristiques remarquables : tout d'abord, de par sa nature, cette méthode tend à concentrer les données près de la moyenne. Ensuite, elle permet une simplicité relative de l'analyse statistique des données obtenues. Cependant, cette méthode présente également trois défauts : premièrement, le test n'a pas de caractéristique d'optimisation mathématique. De plus la précision de l'estimation avec la même taille d'échantillon doit être encore augmentée ou bien la taille de l'échantillon doit être réduite pour la même précision d'estimation. Deuxièmement, la taille du pas de contrainte doit être estimée avant le test. Si la taille de pas est trop grande, la probabilité de générer des données invalides augmente, ce qui accroît l'incertitude de l'essai. Troisièmement, l'estimation de l'écart-type est systématiquement moins précise et inférieure à la valeur réelle. Par conséquent, cette étude a réalisé l'essai de fatigue expérimental par la méthode de l'escalier mais a également amélioré cette méthode en termes de post-traitement et de stratégie d'essai.

F.3. Chapitre 2 : Contrôle expérimental des vibrations basé sur la déformation

Cette étude est une combinaison d'essais expérimentaux de fatigue à amplitude constante avec la technique de contrôle des déformations et la méthode de l'escalier pour évaluer la limite de fatigue dans le domaine des déformations. Le contrôle des déformations permet de réaliser des essais de fatigue en régime permanent jusqu'à la rupture grâce à une jauge de déformation collée au centre de la section réduite de l'éprouvette. Une procédure d'essai en escalier avec un banc de flexion vibrant, comprenant la sélection des paramètres, est détaillée dans ce travail. L'analyse des résultats est basée sur l'estimation de la densité du noyau. Elle est utilisée pour accéder à la limite de fatigue sur une distribution non paramétrique. Des échantillons d'acier à faible teneur en carbone avec une zone de fatigue ont été sélectionnés pour évaluer les caractéristiques statistiques de la limite de fatigue. Les résultats mettent en évidence l'efficacité de la méthode de l'escalier combinée à celle des déformations contrôlées pour atteindre la limite de fatigue.

Dans cette étude, l'agitateur électro-dynamique, refroidi par l'air, est relié à l'éprouvette par une table vibrante. Les expériences sont réalisées sur des éprouvettes en acier F.3. CHAPITRE 2 : CONTRÔLE EXPÉRIMENTAL DES VIBRATIONS BASÉ SUR LA DÉFORMATION à faible teneur en carbone soumises à une excitation cyclique sinusoïdale en déformation par flexion. L'essai de fatigue avec contrôle de déformation est mis en oeuvre avec la technique SRTD [START_REF] Ahmadi | Experimental high cycle fatigue testing and shape optimization of turbine blades[END_REF]. Les conditions de résonance sont maintenus automatiquement tout au long de l'essai même si un changement se produit (fissure, par exemple). Il y a deux caractéristiques dans le SRTD avec contrôle de la déformation. Tout d'abord, il s'agit de contrôler l'amplitude de la déformation pour maintenir une valeur constante. L'agitateur ajuste l'accélération de base pour atteindre le niveau de déformation nécessaire à l'essai de fatigue. Ensuite, le SRTD maintient la différence de phase entre les signaux d'excitation et de réponse à 90 degrés, ce qui garantit que la fréquence d'excitation et la fréquence de résonance de l'échantillon restent les mêmes, car une réduction de la fréquence de résonance est observée lorsqu'une fissure se produit. En général, l'amplitude de déformation forme une boucle fermée dans l'essai SRTD, ce qui garantit une amplitude de déformation constante et le maintien à la fréquence de résonance. Ainsi, le contrôle de la déformation a une réponse très stable dans l'essai SRTD.

Dans l'expérience, le contrôle de la déformation dans le vibrateur pour l'essai de fatigue est validé par la méthode de l'escalier. Le diagramme en escalier de l'essai en escalier est présenté dans la Fig. Pour conclure, les résultats montrent les niveaux de déformation à la valeur pré-définie et maintenue constante pendant l'essai, indépendamment de la variation de la fréquence. Ces données démontrent la fiabilité du contrôle de la déformation lors d'un essai de fatigue par vibration sur l'agitateur électro-dynamique.

F.4. Chapitre 3 : Évaluation de l'incertitude par les méthodes de rééchantillonnage

Le post-traitement donne une estimation précise de la distribution de la limite de fatigue. Cependant, on peut se demander quels seront les résultats si l'essai en escalier est effectué plus d'une fois sur le même système d'essai.

Le Bootstrap est une méthode non paramétrique qui repose sur le rééchantillonnage des échantillons observés. Le point clé du bootstrap est l'échantillonnage non ordonné avec remplacement. Combinée à la méthode de l'escalier, une procédure bootstrap est proposée comme l'Algorithm F.1 : Tout d'abord, une analyse de sensibilité est effectuée pour simuler la variation du facteur d'incertitude en fonction du nombre de spécimens dans un seul échantillonnage et des périodes d'échantillonnage. Ensuite, un total de 100 bootstrap avec 100 échantillons dans chaque échantillonnage est effectué. Nous avons constaté que l'incertitude pour l'estimation de l'écart-type est très élevée, bien plus élevée que l'estimation de la moyenne.

Pour valider la technique de contrôle de la déformation et l'essai en escalier par des méthodes statistiques, trois types de spécimens avec des dimensions différentes ont été utilisés. Sur la base des différents résultats expérimentaux obtenus, le bootstrap est utilisé pour analyser la relation entre les résultats et les causes de leur dispersion. Plusieurs facteurs principaux, notamment la fréquence de résonance et l'épaisseur, sont impliqués dans la dispersion des résultats des tests de fatigue. Ils sont discutés à la fin. Les travaux précédents ont fourni des résultats d'essais expérimentaux en escalier. Ainsi, une méthode de post-traitement est nécessaire pour obtenir la distribution de la limite de fatigue.

Une nouvelle méthode d'évaluation basée sur l'estimation de la densité du noyau (KDE) est proposée pour estimer la distribution de la limite de fatigue à partir d'essais en escalier sans connaissance préalable. Le PDF de tous les échantillons est ensuite estimé par la somme de ces densités du noyau. Sur la base de données expérimentales, l'estimation du KDE avec la règle de Scott et la largeur de bande améliorée conduit à un PDF estimé lisse. Puisque la correction linéaire élargit l'incertitude de l'estimation [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF], le développement d'une approche non linéaire pour réduire le biais de l'écart type est nécessaire.

L'estimation proposée de la limite de fatigue basée sur le KDE avec correction du biais est exprimée par pseudo-code dans l'Algorithm F. Cette méthode combine les informations préalables avec les informations de l'échantillon pour obtenir la probabilité postérieure. La distribution de la limite de fatigue est estimée sur la base de la probabilité postérieure, ce qui réduit l'erreur d'analyse statistique causée par les caractéristiques du petit échantillon de l'essai de limite de fatigue.

Pour le premier spécimen, la priorité initiale est créée à partir de la priorité nulle comme détaillé dans Section 5.2.3. Les contraintes possibles proviennent de la plage de la médiane de distribution des limites de fatigue dans le paramètre d'entrée. La fonction d'utilité, décrite dans Section 5.3.2, est utilisée pour rechercher la contrainte de charge. Après l'essai, l'amplitude de la contrainte et le résultat de l'essai (survie ou échec) sont obtenus, et la postérieure est mise à jour sur la base de ce résultat d'essai. Cette dernière devient l'antérieur pour le spécimen suivant.

Cette procédure est répétée pour les autres échantillons avec BMES jusqu'à ce que la postérieure ait une convergence. Cela signifie qu'une seule distribution peut être estimée à partir de la postérieure. À partir de cette étape, le LHS est appliqué pour déterminer la contrainte de charge et mettre à jour la postérieure. Ensuite, les LHS et BMES sont utilisés alternativement pour déterminer la contrainte de charge. La postérieure est mise à jour après chaque test, quel que soit le LHS ou le BMES. Enfin, la valeur postérieure P (θ|x 1:n ) est obtenue après un total de n spécimens.

La postérieure finale, P (θ|x 1:n ), présente la probabilité des observations compte tenu des paramètres possibles du modèle. L'estimation des paramètres du modèle est obtenue en recherchant le maximum dans la postérieure après le dernier spécimen. Le protocole optimal de l'escalier est donné dans l' Algorithm F.3.

Par rapport à la méthode de l'escalier classique, les avantages de la méthode Bayes-LHS peuvent être énumérés comme suit : En outre, par rapport à la méthode bayésienne en escalier, la méthode Bayes-LHS optimisée fournit un protocole de calcul plus rapide tout en conservant l'exactitude et la précision.

F.7. Chapitre 6 : Conclusion et perspectives

Le présent travail décrit le sujet de l'évaluation des limites de fatigue. La thèse couvre des aspects allant des expériences de fatigue mécanique aux méthodes statistiques. Ce dernier chapitre résume ce que nous avons appris des travaux précédents et indique les directions les plus prometteuses pour les recherches futures.

Selon la revue de la littérature, l'évaluation de la limite de fatigue est un problème pratique difficile car l'éprouvette est exposée à des conditions de vibration, ce qui est crucial pour une conception structurelle fiable. Pour parvenir à une distribution de probabilité de la limite de fatigue, les principaux défis proviennent de la méthode d'essai expérimentale et d'une technique d'évaluation avancée des données d'essai.

L'évaluation de la limite de fatigue commence par l'essai expérimental. Un banc de F.7. CHAPITRE 6 : CONCLUSION ET PERSPECTIVES vibration est une solution raisonnable pour réduire la durée de l'essai de fatigue. En revanche, l'essai de limite de fatigue exige que l'éprouvette soit soumise à une amplitude de contrainte constante pendant un certain nombre de cycles. Une technique de contrôle de la déformation appliquée à un electro-dynamic shaker est proposée pour réaliser un essai à amplitude de déformation constante. L'efficacité de l'approche proposée a été démontrée par des essais de flexion de plaques d'acier DC01 à leur première fréquence de résonance. Le contrôle de la déformation à la résonance (amplification du signal) et l'excitation à relativement haute fréquence permettent de réduire le temps d'essai pour atteindre un grand nombre de cycles. Cette approche est donc efficace pour tester la méthode de l'escalier. Le seuil de fatigue et la dispersion de l'acier DC01 sont estimés par les résultats de l'escalier. Les résultats montrent une bonne répétabilité avec la méthode de l'escalier. La méthode statistique offre des approches permettant de fournir les distributions des limites de fatigue pour chaque essai en escalier. La question clé est : quels sont les résultats d'une nouvelle exécution de l'essai de l'escalier ? Une analyse d'incertitude est effectuée pour évaluer les distributions des limites de fatigue d'un matériau obtenues à partir de l'essai en escalier. Pour cela, nous avons appliqué la méthode de rééchantillonnage, leave-one-out et bootstrap sur les données de l'escalier afin de déduire la dispersion de la moyenne et l'écart type de la distribution. Les résultats de cette étude ont mis en évidence la forte incertitude inhérente à l'estimation de l'écarttype. Les résultats du rééchantillonnage bootstrap montrent que l'écart-type de la limite de fatigue présente une grande dispersion, ce qui permet de conclure qu'il faut être prudent pour utiliser les résultats des escaliers dans la conception de la fatigue.

Pour réduire cette incertitude, nous proposons d'utiliser l'estimation de la densité du noyau (KDE) dans cette étude en raison de son caractère non paramétrique et de son indépendance par rapport au modèle de distribution. Pour la comparer à d'autres méthodes basées sur des hypothèses, la KDE est testée sur différentes distributions afin de valider son efficacité. La dépendance de l'hyperparamètre KDE est également étudiée et optimisée pour améliorer ses performances dans l'évaluation de la limite de fatigue. De plus, comme la méthode de l'escalier nécessite de définir une taille de pas initiale, un facteur corrigé non linéaire est formulé pour réduire son influence dans l'estimation de l'écart-type. L'approche numérique utilise la simulation de Monte-Carlo et permet d'examiner l'effet du nombre de spécimens et de la taille du pas. La performance de l'estimation est évaluée sur la moyenne et l'écart-type de la limite de fatigue impliquant différentes distributions et coefficients de variation.

Dans le but de résoudre la limitation de la taille des marches dans la méthode de l'escalier, un protocole d'escalier Bayes-LHS est présenté pour supprimer la taille des marches dans l'escalier conventionnel. La théorie bayésienne des essais de fatigue et des Bayesian Maximum Entropy Sampling (BMES) est détaillée, puis le protocole d'es-calier bayésien est fourni avec un exemple. Cette étude appelée méthode Bayes-LHS enrichit l'approche Bayes avec les Latin Hypercube Sampling (LHS). Cette proposition est décrite pour la procédure en escalier. Les résultats sont suffisamment optimistes pour considérer que le protocole Bayes-LHS pourrait remplacer la méthode classique de l'escalier pour l'estimation des limites de fatigue.

1 . 1 1 . 2

 1112 Fatigue and fatigue limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Fatigue properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Strain-stress relationship in HCF . . . . . . . . . . . . . . . . . . 6 Staircase test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Staircase test protocol . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Staircase parameters . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. 1 FF. 1 123 A. 1

 111231 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Existing evaluation techniques . . . . . . . . . . . . . . . . . . . . Résumé en français Introduction générale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 Chapitre 1 : Vue d'ensemble de la limite de fatigue . . . . . . . . . . . . F.3 Chapitre 2 : Contrôle expérimental des vibrations basé sur la déformation F.4 Chapitre 3 : Évaluation de l'incertitude par les méthodes de rééchantillonnage166 F.5 Chapitre 4 : Méthode d'évaluation non paramétrique dans un test d'escalier pour la limite de fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.6 Chapitre 5 : Plan d'expérience optimal en escalier par échantillonnage bayésien à entropie maximale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.7 Chapitre 6 : Conclusion et perspectives . . . . . . . . . . . . . . . . . . . Bibliography xiv 4.1 Estimated PDF from different bandwidths (SC: Scott; SI : Silverman; SJ: Sheather-Jones) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 Estimated PDF from different bandwidths . . . . . . . . . . . . . . . . . 79 4.3 Simulation procedure and evaluation for results . . . . . . . . . . . . . . 81 4.4 Normalised (a) median value and (b) standard deviation estimated with Kernel Density Estimation (KDE) for different bandwidth selectors and varying number of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.5 Normalised (a) median value and (b) standard deviation estimated with KDE for different bandwidth selectors and varying normalised step sizes . . . . 86 4.6 Distributions of median obtained from simulation for varying numbers of specimens for (a) d = 1.0 and (b) d = 1.5 normalised . . . . . . . . . . . . . . 87 4.7 Distributions of median obtained from simulation for different normalised step sizes for (a) n = 30 and (b) n = 100 . . . . . . . . . . . . . . . . . . . . . 87 4.8 Distributions of standard deviation obtained from simulation for varying numbers of specimens for (a) d = 1.0 and (b) d = 1.5 normalised . . . . . . . 88 xvi LIST OF FIGURES 4.9 Distributions of standard deviation obtained from simulation for different normalised step sizes for (a) n = 30 and (b) n = 20 . . . . . . . . . . . . . . 89 4.10 Estimated standard deviation and the number of used piecewise equation 90 4.11 Distributions of standard deviation obtained from simulation with the Normal distribution and CV=0.0125 . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.12 Distributions of standard deviation obtained from simulation with the Normal distribution and CV=0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.13 Distributions of standard deviation obtained from simulation with Lognormal distribution and CV=0.025 . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.14 Distributions of standard deviation obtained from simulation with Weibull distribution and CV=0.025 . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.15 PDF of the fatigue limit . . . . . . . . . . . . . . . . . . . . . . . . . . .94 4.16 CDF of the fatigue limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.1 Utility function diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Workflow of the Bayesian staircase procedure . . . . . . . . . . . . . . . 105 5.3 Up-and-down diagram of a simulated Bayesian staircase . . . . . . . . . 106 5.5 Posterior of the Bayesian staircase . . . . . . . . . . . . . . . . . . . . . 106 5.4 Shannon entropy of the Bayesian staircase . . . . . . . . . . . . . . . . . 107 5.6 An example of LHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.7 Workflow of the Bayes-LHS staircase procedure . . . . . . . . . . . . . . 109 5.8 Up-and-down diagram of a simulated Bayes-LHS staircase test . . . . . . 111 5.9 Posterior and estimation of a Bayes-LHS staircase test . . . . . . . . . . 112 5.10 Shannon entropy of a Bayes-LHS staircase test . . . . . . . . . . . . . . . 112 5.11 Simulation procedure and evaluation of the results . . . . . . . . . . . . . 113 5.12 Decrease of the Shannon entropy in case of n = 30 . . . . . . . . . . . . . 114 5.13 Normalised (a) median value and (b) standard deviation with the different median ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.14 Normalised (a) median value and (b) standard deviation with the different standard deviation ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.15 Normalised (a) median value and (b) standard deviation with respect to the number of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.16 Normalised (a) median value and (b) standard deviation with respect to the number of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.17 Normalised (a) median value and (b) standard deviation with respect to step size in conventional staircase method . . . . . . . . . . . . . . . . . . . . 118 6.1 Decrease of strain mean value from Ti-6Al-4V specimens . . . . . . . . . Finite element model of the specimen . . . . . . . . . . . . . . . . . . . . 125 xvii LIST OF FIGURES A.2 A sketch of the clamp in the shaker [76] . . . . . . . . . . . . . . . . . . A.3 Deformed shape and contact pressure of the clamp [76] . . . . . . . . . . A.4 Representation of the clamp in finite element model . . . . . . . . . . . . A.5 The modals of the specimen with thickness 2mm . . . . . . . . . . . . . A.6 FRF of strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7 Harmonic analysis and experiment result . . . . . . . . . . . . . . . . . . A.8 Strain distribution for the first mode . . . . . . . . . . . . . . . . . . . . A.9 Two positions for strain gauge . . . . . . . . . . . . . . . . . . . . . . . . A.10 Stress gradient evolution from the center to the edge (orange line in Fig. A.9)132 A.11 Comparison between the response in the specimen center and on edge obtained by experimental tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.12 Velocity response in specimen center and on the edge of the notch . . . . C.1 Distributions of the median from DM -Number of specimens . . . . . . C.2 Distributions of the median from DM -Normalised step size . . . . . . . C.3 Distributions of standard deviation from DM (corrections) -Number of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Distributions of standard deviation from DM (corrections) -Normalised step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 Left and right censored data distribution and likelihood . . . . . . . . . . D.2 Comparison of estimated PDF from MLE and MLE with censored data . D.3 An example of likelihood function optimization . . . . . . . . . . . . . . D.4 Distributions of median from MLE (Normal) -Number of specimens . . D.5 Distributions of median from MLE (Normal)-Normalised step size . . . D.6 Distributions of standard deviation from MLE (Normal) -Number of specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.7 Distributions of standard deviation from MLE (Normal) -Normalised step size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.8 Distributions of standard deviation from MLE (Lognarmal) . . . . . . . D.9 Distributions of standard deviation from MLE (Weibull) . . . . . . . . . E.1 Normal distribution test . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Lognormal distribution test . . . . . . . . . . . . . . . . . . . . . . . . . E.3 Weibull distribution test . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Courbe S-N et test de l'escalier . . . . . . . . . . . . . . . . . . . . . . . F.2 Diagramme up-and-down pour les échantillons d'acier DC01 . . . . . . . xviii List of Tables 1.1 Comparison of test protocols for fatigue limit distribution . . . . . . . . 1.2 Summary of staircase test and fatigue limit characteristics . . . . . . . . 2.1 Difference of fatigue test machines . . . . . . . . . . . . . . . . . . . . . 2.2 Mechanical properties of the low carbon steel DC01 . . . . . . . . . . . . 2.3 Chemical composition of the steel DC01 material . . . . . . . . . . . . . 2.4 Configurations of the shaker . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The selected specimens for fractography analysis . . . . . . . . . . . . . . 2.6 Empirical Coefficients b i (i = 1, 2, 3, 4) for K owen . . . . . . . . . . . . . . 2.7 Empirical Coefficients c i (i = 1, 2, 3) for K owen . . . . . . . . . . . . . . . 2.8 The experimental parameters used in staircase test . . . . . . . . . . . . 2.9 The result data of specimens with thickness 2mm . . . . . . . . . . . . . 2.10 Fatigue limit values with specified P-C levels . . . . . . . . . . . . . . . 2.11 Comparison of fatigue limits of carbon steel . . . . . . . . . . . . . . . . 3.1 Effect of dimensions on the resonant frequency . . . . . . . . . . . . . . . 3.2 Comparison of median and standard deviation (with DM) . . . . . . . . 4.1 Overview of existing evaluation techniques for staircase testing . . . . . . 4.2 The coefficients used in the correction equation (Eq. 4.13) . . . . . . . . 4.3 Parameters in simulation work . . . . . . . . . . . . . . . . . . . . . . . . 4.4 J-S divergence with different α parameter (×10 -2 ) . . . . . . . . . . . . . 4.5 The J-S divergence of all the methods investigated in the study cases with d=1.0 and n=30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The median and standard deviation from different estimation . . . . . . 5.1 Different input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Different treatments for Ti-6Al-4V specimens used in Fig. 6.1 . . . . . . A.1 The first resonant frequencies estimated for several boundary conditions . B.1 Effect of dimensions on the resonant frequency . . . . . . . . . . . . . . . B.2 The specimen, test data and test results from staircase test of specimen S1 B.3 The specimen, test data and test results from staircase test of specimen S2L137 xix LIST OF TABLES C.1 Constants used in standard deviation correction (Eq. C.6) . . . . . . . . 141 xx Glossaries C : confidence level E : Young's modulus H : Shannon entropy I : Shannon information K f : fatigue notch factor K t : stress concentration factor M : number of resampling replications N : the number of specimens in a single sampling N L : number of cycles fixed for defining fatigue limit N f : fatigue life (number of cycles to failure) P : probability P f : probability of failure S a : load amplitude in staircase test S e : endurance limit (MPa) S f : fatigue limit (MPa) S a,0 : lowest stress level in staircase test (MPa) U : utility function θ : distribution model parameters δ m : uncertainty factor of median δ s : uncertainty factor of standard deviation σ f : fatigue strength coefficient σ u : ultimate strength σ y : yield strength σ max : maximum stress value σ min : minimum stress value ε : strain (µm/m) ε f : fatigue ductility coefficient d : step size in staircase test GLOSSARIES f r : resonant frequency (Hz) h : KDE bandwidth i : ordinal number of specimens j : ordinal number of load levels m : median value of samples n : total number of specimens or sample size in staircase test n f,j : number of failures at load level j n s,j : number of survivals at load level j s : standard deviation of samples xxii BMES Bayesian Maximum Entropy Sampling CDF Cumulative Density Function CRF Change of Resonant Frequency DM Dixon-Mood DOF Degree Of Freedom EDM Electrical Discharge Machine FEA Finite Element Analysis FEM Finite Element Method FRF Frequency Response Function HCF High Cycle Fatigue KDE Kernel Density Estimation LCF Low Cycle Fatigue LHS Latin Hypercube Sampling LMN Laboratory of Mechanics of Normandy Loo Leave-one-out Loocv Leave-one-out cross-validation MCS Monte-Carlo Simulation MLE Maximum Likelihood Estimation MSE Mean Square Error PDF Probability Density Function SCF Stress Concentration Factor SEM Scanning Electron Microscopy SRTD Sine Resonance Track-and-Dwell VHCF Very High Cycle Fatigue xxiii Chapter 1

Figure 1

 1 Figure 1.1: S-N curve illustration

Figure 1 . 2 :

 12 Figure 1.2: S-N curve and the fatigue limit

Figure 1 . 3 :

 13 Figure 1.3: Distribution representing the probabilistic nature of the fatigue tests

1. 1 . 2 stress amplitude σ a = ∆σ 2

 122 FATIGUE AND FATIGUE LIMIT σ min and σ max : stress range ∆σ = σ max -σ min mean stress σ m = σmax+σ min

Figure 1 . 4 :

 14 Figure 1.4: Stress-strain relationship

Figure 1 . 5 :

 15 Figure 1.5: Typical plot of strain amplitude versus cycles to failure

Figure 1 . 6 :

 16 Figure 1.6: S-N curve and the staircase test

Figure 1 . 8 :

 18 Figure 1.8: Framework of this study

CHAPTER 2 .

 2 EXPERIMENTAL STRAIN-BASED VIBRATION CONTROL TO OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD

  2.1.

Figure 2 . 1 :

 21 Figure 2.1: Geometry and dimension of the symmetrical specimen (Unit: mm)

Figure 2 . 3 :

 23 Figure 2.3: The first four mode shapes of the specimen. (The colors correspond to the normalized displacement. Red is the maximum)

Figure 2 . 4 :

 24 Figure 2.4: FRF of strain response

Figure 2 . 5 :Figure 2 . 6 :

 2526 Figure 2.5: Schematic of the testing setup

Figure 2 . 7 :

 27 Figure 2.7: Dimension of strain gauge (HBM 1-LM11-1.5/350GE)

Figure 2 . 8 :

 28 Figure 2.8: FRF of the strain and the velocity to the base acceleration

Figure 2 . 9 :

 29 Figure 2.9: Strain and velocity vs. the acceleration level

Figure 2 . 10 :

 210 Figure 2.10: Schema of the change of the bench instrumentation

29 CHAPTER 2 .Figure 2 . 11 :

 292211 Figure 2.11: SRTD test with strain control

  fr number of cycles, N Define strain, ε initial frequency, fr0, number of cycles, NL DAQ Set resonant frequency, fr

Figure 2 . 12 :

 212 Figure 2.12: Test control and condition system for SRTD test

Figure 2 . 13 :

 213 Figure 2.13: Fatigue test procedure with strain control

Figure 2 . 14 :

 214 Figure 2.14: Crack length versus resonant frequency decrease of (a) 2% (b) 5% and (c) 8%

Figure 2 . 15 :

 215 Figure 2.15: Comparison of sine sweep test with acceleration control and strain control

Figure 2 . 16 :

 216 Figure 2.16: Comparison of SRTD test with acceleration control or strain control

Fig. 2 .

 2 17 presents the landscapes of the fracture surfaces of strain control and CHAPTER 2. EXPERIMENTAL STRAIN-BASED VIBRATION CONTROL TO OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD

Figure 2 . 17 :

 217 Figure 2.17: Landscape of the fracture surface

  (a) corresponds more to a tensile fracture surface. This observation would require further investigation. Intergranular fracture (a) Strain control Intergranular fracture (b) Acceleration control

Figure 2 . 18 :

 218 Figure 2.18: Direct observation of fracture surface by SEM

Figure 2 . 19 :

 219 Figure 2.19: Experimental data and the related Basquin curve with strain control

Figure 2 . 20 :

 220 Figure 2.20: Staircase test procedure

CHAPTER 2 .

 2 EXPERIMENTAL STRAIN-BASED VIBRATION CONTROL TO OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD

Figure 2 .

 2 Figure 2.21: Up-and-down diagram for DC01 steel specimens

  (a) Frequency decrease (normalized with the initial resonant frequency) (b) Strain amplitude

Figure 2 . 22 :

 222 Figure 2.22: Frequency decrease and strain amplitude during SRTD test. The failure curves correspond to failed specimen ordinal i

Figure 2 . 23 :

 223 Figure 2.23: The acceleration and velocity in same strain level(1380 µm/m)

Figure 2 . 24 :

 224 Figure 2.24: Resonant frequency decrease progress of No. 22 specimen

Figure 2 . 25 :

 225 Figure 2.25: Comparison about cracks and CRF

  2.21), the Probability Density Function (PDF) and Cumulative Density Function (CDF) of fatigue limits calculated by the DM methods are shown in Fig. 2.26 and Fig. 2.27.

Figure 2 .

 2 Figure 2.26: PDF estimated from DM method

Figure 2 . 28 :

 228 Figure 2.28:The CDF of C20 steel[START_REF] Delahay | Estimation of the fatigue strength distribution in high-cycle multiaxial fatigue taking into account the stress-strain gradient effect[END_REF]

Figure 3 . 1 :

 31 Figure 3.1: PDFs of fatigue strain limits from Loo re-sampling

8 9 foreach j in N do 10 14 l

 91014 Reset staircase dataset X = [x 1 , x 2 , ..., x n ] Sample a data x from X l randomly 11 if the sampled specimen x is failure then12 l = l -d //the next sample is selected from specimens in lower level 13 else = l + d // the next sample is selected from specimens in upper level 15 end if 16 get all test point in new level X l17 end foreach 18 get sampling results: X = [x 1 , x 2 , ..., x N ] 19 estimate fatigue limit distribution by the same post-processing method 20 end foreach 21 get all sampling results: [ X 1 , X 2 , ..., X M ] and corresponding uncertainty factors. the bootstrap algorithm. Therefore, a sensitivity analysis is carried out to simulate the change in the uncertainty factor with M and N in the range of 10 to 200. The proposed analysis is implemented in Python. To avoid the potential influence of the number of seeds used in the random process, the proposed procedure is run by 5 times with different numbers of seeds. All 5 runs converge to the same result, indicating that the proposed procedure is independent of the random or numerical configurations. A typical result on the change of M and N are shown in the form of heatmaps in Fig. 3.2 and Fig. 3.3.

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Uncertainty factor of the median for different values of M and N

Figure 3 . 4 :Figure 3 . 5 :

 3435 Figure 3.4: PDF of bootstrap resampling based on the experimental data

20 Figure 3 . 6 :

 2036 Figure 3.6: Shape of the specimen with the same reduced section (Unit: mm )

Figure 3 . 7 :Figure 3 . 8 :Figure 3 . 9 :

 373839 Figure 3.7: Up-and-down diagram for specimen S1

3. 4 .Figure 3 . 11 :Figure 3 . 12 :

 4311312 Figure 3.11: Bootstrap sampling applied to the three kinds of specimens

CHAPTER 4 .

 4 A NON-PARAMETRIC EVALUATION METHOD IN STAIRCASE TEST FOR IMPROVING FATIGUE LIMIT ASSESSMENT 4.1. Background 4.1.1

. 1 .

 1 Figure 4.1: Estimated PDF from different bandwidths (SC: Scott; SI : Silverman; SJ: Sheather-Jones)

Table 4 . 2 :

 42 The coefficients used in the correction equation (Eq. 4.13) respect to different values of CV, like Fig.4.10. In this study, the coefficient of variation (CV) is expressed as the ratio of the standard deviation to the mean:

Algorithm 4 . 1 :1

 41 Non-parametric fatigue limit estimation method Data: The staircase dataset: X = {x 1 , x 2 , ..., x n }, the step size: d Calculate the bandwidth h by Scott's rule; 2 (m, s) from KDE with bandwidth h ; 3 Correction to s ; Result: The estimated median m and the estimated standard deviation s of the fatigue limit distribution 4.3. Numerical validation of the proposed evaluation method 4.3.1. Simulation procedure

4. 3 .Figure 4 . 3 :

 343 Figure 4.3: Simulation procedure and evaluation for results

  in this study, which means that the tests are only valid after the first pair of tests with opposite results.. Thirdly, all results obtained from the staircase simulation are post-processed by three different evaluation techniques, including KDE, MLE and DM, as compared on the right side of Fig. 4.3. Each post-processing method estimates the median value m k , standard deviation s k and probability distribution for the k th staircase test with k = 1, • • • , N .

Fig. 4 .

 4 [START_REF] Liu | Fatigue limit prediction of notched plates using the zero-point effective notch stress method[END_REF] shows the median and standard deviation estimations for the different numbers of specimens. Here, the step size is arbitrarily chosen equal to the true standard deviation d = s 0 . Similarly, Fig.4.4 and Fig.4.5 show the median and standard deviation estimates for different step sizes with a number of specimens equal to 30.

Figure 4 . 4 :

 44 Figure 4.4: Normalised (a) median value and (b) standard deviation estimated with KDE for different bandwidth selectors and varying number of specimens

  Fig. 4.6 shows the influence of the number of specimens for d = 1.0 and d = 1.5 normalised, and Fig. 4.7 shows the effect of the step size for a small number of specimens (n = 30) and a larger one (n = 100).

Figure 4 . 6 :

 46 Figure 4.6: Distributions of median obtained from simulation for varying numbers of specimens for (a) d = 1.0 and (b) d = 1.5 normalised

Figure 4 . 7 :

 47 Figure 4.7: Distributions of median obtained from simulation for different normalised step sizes for (a) n = 30 and (b) n = 100

4. 3 . 5 .

 35 Comparison of the standard deviation estimation 4.3.5.1. Comparison of standard deviations with the Normal distribution (case 1)

Figure 4 . 8 :Figure 4 . 9 :

 4849 Figure 4.8: Distributions of standard deviation obtained from simulation for varying numbers of specimens for (a) d = 1.0 and (b) d = 1.5 normalised

Figure 4 . 10 :

 410 Figure 4.10: Estimated standard deviation and the number of used piecewise equation

Figure 4 . 11 :Figure 4 . 12 :

 411412 Figure 4.11: Distributions of standard deviation obtained from simulation with the Normal distribution and CV=0.0125

Figure 4 . 13 :Figure 4 . 14 :

 413414 Figure 4.13: Distributions of standard deviation obtained from simulation with Lognormal distribution and CV=0.025

Figure 4 .

 4 Figure 4.15: PDF of the fatigue limit

Figure 4 .

 4 Figure 4.16: CDF of the fatigue limit

Figure 5 . 1 :

 51 Figure 5.1: Utility function diagram

Figure 5 . 2 :

 52 Figure 5.2: Workflow of the Bayesian staircase procedure

5. 4 .Figure 5 . 4 :

 454 Figure 5.4: Shannon entropy of the Bayesian staircase

  Fig. 5.6(b).

Figure 5 . 6 :

 56 Figure 5.6: An example of LHS

( 3 )

 3 The possible range of the standard deviation of the fatigue limit distribution. The selection of the standard deviation range should come from the references and rely on the engineering experience. REPRESENTATION The input parameters under the Normal distribution can be expressed as: m s = m 1 , m 2 , . . . , m a s 1 , s 2 , . . . , s b (5.23)

Figure 5 . 7 :

 57 Figure 5.7: Workflow of the Bayes-LHS staircase procedure

Figure 5 . 8 :

 58 Figure 5.8: Up-and-down diagram of a simulated Bayes-LHS staircase test

Figure 5 . 9 :Figure 5 . 10 :

 59510 Figure 5.9: Posterior and estimation of a Bayes-LHS staircase test

Figure 5 . 11 :

 511 Figure 5.11: Simulation procedure and evaluation of the results

Figure 5 . 12 :

 512 Figure 5.12: Decrease of the Shannon entropy in case of n = 30

114 5. 7 .

 1147 RESULTS AND DISCUSSIONS

Figure 5 . 13 :Figure 5 . 14 :

 513514 Figure 5.13: Normalised (a) median value and (b) standard deviation with the different median ranges

Figure 5 . 15 :

 515 Figure 5.15: Normalised (a) median value and (b) standard deviation with respect to the number of specimens

  .

Figure 5 . 16 :

 516 Figure 5.16: Normalised (a) median value and (b) standard deviation with respect to the number of specimens

Figure 5 . 17 :

 517 Figure 5.17: Normalised (a) median value and (b) standard deviation with respect to step size in conventional staircase method

  As shown inFig. A.2, the experimental clamping is intended to ensure a high rigidity in the translate direction of DX, DY and DZ. Also, the rotation DRZ is fixed by two plates. DRY is fixed by the lower plate, DRX is also fixed because the mass of specimens is much lower than the vibratory bench (not directly connected with the shaker armature) [76]. The boundary condition of all 6 DOFs in FE model is marked in Fig. A.4.

Figure A. 4 :

 4 Figure A.4: Representation of the clamp in finite element model

  Fig. A.8.

Figure A. 8 :Figure A. 9 :

 89 Figure A.8: Strain distribution for the first mode

  Fig. A.10.

Figure A. 11 :

 11 Figure A.11: Comparison between the response in the specimen center and on edge obtained by experimental tests

Figure C. 3 :Figure C. 4 :

 34 Figure C.3: Distributions of standard deviation from DM (corrections) -Number of specimens

3 )

 3 All methods including DM and DM with bootstrap give almost the same and accurate estimation for the mean. 4) From Fig. C.4, the SL correction indeed improves original DM especially for N = 30. 5) PO correction include the parameter n and d, so it provides the estimations closer to real value. However, it leads to large diversity and is not easy to use due to the un-constant parameters. 6) From Fig. C.3, and Fig. C.4(a), The BZ correction appears not to better results[38],

  An example is given as: a staircase including 30 specimens with initial distribution N (400, 10). The estimated results of classical MLE and MLE with censoring are listed in Fig. D.2.

Figure D. 2 :

 2 Figure D.2: Comparison of estimated PDF from MLE and MLE with censored data

Figure D. 4 :Figure D. 5 :

 45 Figure D.4: Distributions of median from MLE (Normal) -Number of specimens

Figure D. 6 :Figure D. 7 :

 67 Figure D.6: Distributions of standard deviation from MLE (Normal) -Number of specimens

4 )Figure D. 8 :Figure D. 9 :

 489 Figure D.8: Distributions of standard deviation from MLE (Lognarmal)

F. 1

 1

Figure F. 1 :

 1 Figure F.1 : Courbe S-N et test de l'escalier

F. 2 .

 2

Figure F. 2 :

 2 Figure F.2 : Diagramme up-and-down pour les échantillons d'acier DC01

F. 5 . 9 foreach j in N do 10 Echantillonner 14 l

 591014 CHAPITRE 4 : MÉTHODE D'ÉVALUATION NON PARAMÉTRIQUE DANS UN TEST D'ESCALIER POUR LA LIMITE DE FATIGUE Algorithm F.1: Bootstrap 1 L'ensemble de données en escalier qui comprend n simples est représenté par X = [x 1 , x 2 , ..., x n ], et la taille de l'étape est d 2 Définir N comme le nombre d'échantillons dans un seul échantillonnage et M comme les durées d'échantillonnage 3 Déterminer tous les niveaux de déformation (ou de contrainte)L = [l 1 , l 2 , ...l]4 Diviser X en différentes parties selon les différents niveaux de déformation (ou de contrainte) l, soitX l = [x l 1 , x l 2 , ..., x l ] , c'est-à-dire X 1 ∪ X 2 ∪ ...X l = X 5 Pré-requis : Sélectionner aléatoirement un niveau de souche d'essai l 6 Obtenir toutes les données de test du niveau de souche l 7 foreach i in M do 8 Réinitialiser l'ensemble de données en escalier X = [x 1 , x 2 , ..., x n ] une donnée x à partir de X l de façon aléatoire 11 if l'échantillon x est défaillant then 12 l = l -d // l'échantillon suivant est sélectionné parmi les spécimens du niveau inférieur 13 else = l + d // l'échantillon suivant est sélectionné parmi les spécimens du niveau supérieur 15 end if 16 Obtenir tous les points de test dans le nouveau niveau X l 17 end foreach 18 Obtenir les résultats de l'échantillonnage : X = [x 1 , x 2 , ..., x N ] 19 Estimer la distribution de la limite de fatigue par la même méthode de post-traitement 20 end foreach 21 Obtenir tous les résultats de l'échantillonnage : [ X 1 , X 2 , ..., X M ] et les facteurs d'incertitude correspondants F.5. Chapitre 4 : Méthode d'évaluation non paramétrique dans un test d'escalier pour la limite de fatigue

2 .Algorithm F. 2 :1F. 6 .

 226 Méthode non paramétrique d'estimation de la limite de fatigue Data: L'ensemble de données sur les escaliers : X = {x 1 , x 2 , ..., x n }, la taille de l'escalier : d Calcul de la largeur de bande h par la règle de Scott; 2 (m, s) de KDE avec une largeur de bande h ; 3 Correction de s ;Result: La médiane estimée m et l'écart type estimé s de la distribution de la limite de fatigue Une simulation numérique modélisant l'essai en escalier est réalisée avec différentes distributions et coefficients de variation. La valeur médiane (m) et l'écart-type (s) sont utilisés dans ce travail pour décrire la distribution des limites de fatigue. Il apparaît que la méthode KDE a un résultat estimé conforme à la méthode MLE. La médiane et l'écart-type des estimations de KDE et MLE convergent les uns vers les autres. En comparant les distributions estimées par KDE, MLE et DM, les principaux résultats peuvent être tirés :(1) : MLE distingue les spécimens en fonction du résultat de l'échec ou de la survie, tandis que KDE ne requière pas cette information (2) KDE a un résultat estimé selon la méthode MLE. La médiane et l'écart-type des estimations de KDE et MLE convergent les uns vers les autres (3) DM donne un plus grand écart-type de la limite de fatigue. Alors qu'après la correction de l'écart-type dans la méthode KDE, la valeur prédite devient beaucoup plus petite Pöur conclure, la méthode KDE proposée réduit considérablement le biais de l'écart-type de la limite de fatigue à partir d'essais sur petits échantillons. De plus, les performances de cette méthode sont insensibles aux paramètres d'essai de la méthode de l'escalier. F.6. CHAPITRE 5 : PLAN D'EXPÉRIENCE OPTIMAL EN ESCALIER PAR ÉCHANTILLONNAGE BAYÉSIEN À ENTROPIE MAXIMALE Chapitre 5 : Plan d'expérience optimal en escalier par échantillonnage bayésien à entropie maximale Pour remédier aux défauts intrinsèques de la méthode d'essai conventionnelle en escalier, ce chapitre propose une méthode d'essai optimisée, basée sur les Echantillonages Bayésiens à Entropie Maximale (BMES) et les Echantillonages Latins Hypercubes (LHS) pour déterminer la distribution de la limite de fatigue de la structure.

( 1 )Algorithm F. 3 : 4

 134 Cette méthode évite l'utilisation d'une contrainte de départ devinée et d'une taille de pas fixe. Elle sélectionne artificiellement la gamme des paramètres possibles du modèle avant le test bayésien en escalier. Framework of Bayes-LHS staircase protocol Data: Distribution ; plages de la médiane et de l'écart-type ; 1 Générer le paramètre de modèle θ. Initialisation de l'antériorité P (θ) ; if P (θ|x 1:i ) convergence then 2 Déterminer l'amplitude de la contrainte de charge par BMES 3 else Déterminer l'amplitude de la contrainte de charge par LHS et BMES alternativement 5 end if 6 Tester l'éprouvette 7 Calculer la postérieure P (θ|x 1:n ) et l'entropie de Shannon après chaque test 8 Postérieure finale Result: La médiane estimée m et l'écart type estimé s de la distribution de la limite de fatigue (2) L'utilisation d'informations antérieures pendant le test permet une convergence plus rapide [116], ce qui signifie moins de spécimens pour atteindre les mêmes résultats. (3) Très peu d'informations sont nécessaires sur le jeu d'échantillons avant le test. (4) Il n'est pas nécessaire d'utiliser une méthode de post-traitement. Le paramètre du modèle de distribution est obtenu directement à partir du modèle postérieur.

  2.1 Fatigue test machines and their control . . . . . . . . . . . . . . . . . . . 2.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variability in the experimental test . . . . . . . . . . . . . . . . . 2.2.4 Linearity of the test system in acceleration control . . . . . . . . . 2.3 Strain control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Experimental development for fatigue tests . . . . . . . . . . . . . 2.3.2 Strain gauge and failure criterion . . . . . . . . . . . . . . . . . . 2.3.3 Comparative study of fatigue test with strain control and acceleration control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Fatigue curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Application of staircase method for the fatigue limit estimation . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Application of the leave-one-out resampling . . . . . . . . . . . . . . . . 3.3 Bootstrap resampling and results . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Experiment-based hyper-parameter sensitivity analysis . . . . . . 3.3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Effect of specimen dimensions . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Staircase tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Application of the bootstrap sampling . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	CONTENTS
	2.5 Chapter 3	Uncertainty quantification using resampling
	methods
	3.1
		xi

2.2.1 Specimen design and material . . . . . . . . . . . . . . . . . . . . 2.2.2 Test bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 2.4.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Test results and discussion . . . . . . . . . . . . . . . . . . . . . .

Table 1 . 1 :

 11 Comparison of test protocols for fatigue limit distribution

	Method	Number of specimens	Stress amplitude Evaluation Remarks
	Staircase	multi	one stress per sample	distribution -most widely used
	Step stress	multi	multi stresses per sample	distribution -much more test times than staircase
	Probit	multi	one stress per sample	distribution -much more test specimens than staircase
	Locati	single	multi stresses per sample	mean value	-accelerated test time -need prior fatigue parameter (e.g. the slope of the S-N curve)
	Prot	multi	multi stresses per sample	mean value	-statistical efficiency is less than that of probit and staircase methods [33]

Table 1 . 2 :

 12 Summary of staircase test and fatigue limit characteristics

	Materiel	N L	Stress range (MPa)	d (MPa)	n Statistical Estimation Ref
	Lthium disilicate glass-ceramic	2.5 × 10 5	790-940N	50N	15	DM	m= 879.28 [48]
	EN-GJS 700 ductile cast iron	2 × 10 6	190-230	18.88	7	DM	m= 210.90 s= 10.01	[49]
	SAE 4340 steel	1 × 10 7	306.8-320.6	3.45	54	DM	m= 319.02 s= 20.03	[50]
	Ti-6Al-4V	1 × 10 7	460-590	25	27	DM	m= 540 s= 60	[51]
	Alloy A319 (120℃) 1 × 10 8	68.5-82.8	2.8	26	MLE (censoring)	m= 75 s= 4.2	[52]
	Nodular cast iron	-	177.3-323.1	18.3	25	MLE (censoring)	m= 195.5 s= 17.6	[41]
	Stainless steel	1 × 10 8	205-225	5	22	DM	m= 3.070 s= 3.2e-3	[47]
	C45 steel	5 × 10 6	288-318	6	20	DM	m= 310.3 s= 11.4	[46]
	Ti-6Al-4V	1 × 10 9	400-440	20	18	DM	m= 406.36 s= 11.65	[38]
	EA4T axle steel	-	345-370	5	17	DM	m= 356 s= 13.4596	[24]
	35NCDV12 steel Non-chromium	5 × 10 6	388-529	47	15	DM	m= 393.875 s= 39.088	[53]
	35NCDV12 steel Chromium	5 × 10 6	151-341	115	15	DM	m= 239.214 s= 50.35	[53]
	ASTM A743 CA6NM steel	2 × 10 6	337-392	14	15	DM	m= 361.625 s= 11.589	[54]
	CL65 steel	-	410-420	5	15	DM	m= 414.64 s= 2.65	[24]
	LZ50 axle steel	-	255-285	10	14	DM	m= 267.14 s= 5.3	[24]
	High strength steel 300 M	1 × 10 9	690-790	20	14	Null	m=723±22 [55]
	Aluminum A7N01S-T4	1 × 10 7	125-140	5	12	MLE fit	Lognormal s= 2.469e-2 m= 2.129	[56]
	Bogie cast steel	-	134-146	4	12	DM	m= 139.23 s= 3.2479	[47]
	Aluminium 7075 T7531	1 × 10 7	187-208	7	11	DM	m= 189 s= 3.7	[57]

Table 2 . 1 :

 21 Difference of fatigue test machines

		Servo-hydraulic machine	Electro-dynamic shaker	Ultrasonic fatigue tester
	Frequency	low (⩽30Hz)	high (5 -7000Hz)	very high (20 -30kHz)
	Test time	long test time	large reduction of test time	large reduction of test time
	Load	displacement or force	acceleration, velocity, displacement, force (with prestress)	displacement (strain)
	Response	well-controlled	variations	well-controlled
				frequency effect
	Drawbacks	low frequency	indirect control	thermal effect
				specific specimen

Table 2 . 2 :

 22 Mechanical properties of the low carbon steel DC01

		Density (kg/m 3 )	Young's modulus (GPa)	Poisson's ratio	Yield Strength (MPa)
			7850	205		0.3		350 [61]
		Table 2.3: Chemical composition of the steel DC01 material
	Element	C	Mn	Si	P	S	Ni	Cr	Al	Mo	N
	Wt%	0.019 0.160 0.008 0.015 0.019 0.019 0.021 0.030 0.003 0.0024
	A modal analysis by Finite Element Method (FEM) of the specimen used in exper-
	imental tests is performed using Ansys. The detailed analysis procedure is presented
	in Appendix A. Fig. 2.2 illustrates the finite element model and measure points. This
	analysis shows the first four mode shapes in Fig. 2.3 with the related normal displace-
	ments representing the deformations and the resonant frequencies.

  .6. CHAPTER 2. EXPERIMENTAL STRAIN-BASED VIBRATION CONTROL TO OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD

Table 2 . 4 :

 24 Configurations of the shaker

	Mass of mobile table	21.0 kg
	Frequency range	5 Hz -2000 Hz
	Force in sine mode	3000 N (p)
	Force in random mode (ISO-5344[75]) 3000 N (eff)
	Acceleration sine	14.6 g (p)
	Acceleration random (ISO-5344[75])	14.6 g (eff)
	Velocity	2 m/s (p)
	Displacement	25 mm (pp)
	(eff: effective value, p:peak value, pp:peak-peak value)

Table 2 . 5 :

 25 The selected specimens for fractography analysis

		Specimen 1	Specimen 2
	Control	Strain control 1360 µm/m	Acceleration control 5 g
	Resonant frequency (Hz)	203.27 Hz	202.65 Hz
	CRF(%)	15.76	25
	Cycles (×10 5 )	6.86078	7.24319

CRF: Change of resonant frequency Eq. 2.1

Table 2 . 6 :

 26 Empirical Coefficients b i (i = 1, 2, 3, 4) for K owen

	Confidence level(C)	b 1	b 2	b 3	b 4
	0.95	0.9968 0.1596 0.60 -2.636
	0.90	1.0030 -6.0160 3.00 1.099
	0.85	1.0010 -0.7212 1.50 -1.486
	0.80	1.0010 -0.6370 1.25 -1.554
	Ref: Page 116 in [86]

Table 2 . 7 :

 27 Empirical Coefficients c i

Table 2 . 8 :

 28 The experimental parameters used in staircase test

	Resonance mode	First bending
	Step size (d)	20 µm/m
	Stop criterion	CRF= 5%
	Number of cycles (N L )	1 × 10 6

Table 2 .

 2 9: The result data of specimens with thickness 2mm

	Number of	cracks	1	2	1	0	0	0	2	1	0	2	0	0	2	2	1	0	1	0	2	0	2	2	0	2	0	0	1	2	2	0	2	2	0	1	0
	End of the test	N end (×10 5 ) (%CRF )	10.64723 7.21	9.28761 11.65	8.76143 4.87	11.14228 4.05	10.9229 0.58	10.97323 1.4	8.911 10	5.60119 5.88	9.99639 0.93	8.716 8.32	11.07563 1.11	10.76577 0.65	7.06643 8.11	10.01248 5.47	7.57889 6.97	10.93247 0.62	9.98718 7.27	11.14542 0.42	10.73098 6.19	10.75909 1.43	9.61367 17.3	9.22806 7.3	10.81194 0.64	8.99245 11.2	10.8827 0.52	10.78389 1.02	6.45991 9.3	9.81772 6.75	6.86078 15.76	10.88091 0.36	7.72214 5.37	9.09296 16.48	10.96331 0.34	10.94223 4.14	11.95499 1.02	8.22478 2.74
	Result	(-)	survival	failure	failure	survival	survival	survival	failure	failure	survival	failure	survival	survival	failure	failure	failure	survival	failure	survival	survival	survival	failure	failure	survival	failure	survival	survival	failure	failure	failure	survival	failure	failure	survival	survival	survival	failure
	Number of cycles (N, ×10 5 )	%CRF = 1 %CRF = 2 %CRF = 3 %CRF = 4 %CRF = 5	9.32805 10.21816 10.49385 10.58351 10.61749	7.98029 8.82029 9.0683 9.16992 9.22115	7.64409 8.49476 8.68159 8.74068 -	9.32047 10.49341 10.92247 11.13286 -	-----	10.07389 ----	4.12443 5.37319 6.16511 6.69891 7.1896	4.7849 5.37969 5.52876 5.57474 5.59181	8.8232 ----	7.56417 8.35942 8.55948 8.64134 8.67903	10.78161 ----	-----	6.25252 6.86953 6.99857 7.03288 7.04796	8.20261 9.23962 9.64369 9.83689 9.96179	6.48485 7.27884 7.47429 7.53749 7.56022	-----	8.7732 9.66207 9.87187 9.93853 9.96492	-----	8.92698 10.12819 10.47181 10.58444 10.66189	10.19047 ----	8.61541 9.28369 9.45339 9.51826 9.55036	7.94693 8.87729 9.09728 9.17367 9.20202	-----	8.44418 8.76087 8.88198 8.93144 8.95403	-----	10.74276 ----	5.63142 6.18471 6.34862 6.40395 6.4285	7.59175 8.58759 9.09189 9.39481 9.59652	5.17888 5.84764 6.06433 6.19833 6.29496	-----	6.94097 7.446 7.61189 7.68253 7.71464	7.54535 8.41179 8.74391 8.88915 8.95725	-----	8.01821 9.39906 10.22085 10.86252 -	11.06726 ----	7.09398 8.04198 ---
	Strain f r 0	(µm/m) (Hz)	1360 207.41	1380 206.51	1360 205.25	1340 208.1	1360 206.51	1380 205.76	1400 209.85	1380 207.3	1360 206.76	1380 204.04	1360 208.01	1380 205.64	1400 207.19	1380 205.05	1360 208.21	1340 207.79	1360 208.05	1340 207.13	1360 207.66	1380 206.4	1400 207.47	1380 209.61	1360 201.31	1380 204.91	1360 202.5	1380 200.83	1400 203.52	1380 201.28	1360 203.27	1340 202.59	1360 203.64	1340 203.85	1320 203.75	1340 204.76	1360 204.57	1380 203.83
	Order Mass	i (g)	1 27.555	2 27.513	3 27.17	4 27.486	5 27.517	6 27.482	7 27.469	8 27.564	9 27.509	10 27.257	11 27.584	12 27.255	13 27.514	14 27.52	15 27.479	16 27.643	17 27.456	18 27.233	19 27.478	20 27.466	21 27.331	22 27.494	23 27.391	24 27.605	25 27.617	26 27.224	27 27.477	28 27.579	29 27.319	30 27.461	31 27.619	32 27.568	33 27.508	34 27.611	35 27.493	36 27.626

1 48 2.4. APPLICATION OF STAIRCASE METHOD FOR THE FATIGUE LIMIT ESTIMATION

  OBTAIN THE FATIGUE LIMIT BY THE STAIRCASE METHOD tolerance limit for a normal distribution with point P . µ P and µ γ are the standard Normal distribution values with the probability P and γ, respectively. Based on the µ =1366.67 µm/m and s =23.44 µm/m, the fatigue limit values with specified P ˘C levels of DC01 steel by the previous approaches are exhibited in Tab. 2.10. Comparison to the steel fatigue limit in the literature The Comparison of some fatigue limit values of carbon steel in fully reversed cycling from the literature is summarised in Tab. 2.11.

		Table 2.10: Fatigue limit values with specified P-C levels
	C	P	0.5	0.9	0.95	0.99	0.999	0.9999
	0.5	1366.67 1336.63 1328.11 1312.14 1294.23 1279.49
	0.9	1361.60 1328.99 1319.30 1300.89 1280.05 1262.82
	0.95 1360.11 1326.44 1316.31 1297.01 1275.12 1257.00
	0.99 1357.21 1321.02 1309.90 1288.61 1264.37 1244.27
					Unit: µm/m
	2.4.2.5.

1 -µγ 2 2(n-1)

(2.11)

where P means survival probability, k(P, 1 -C, n) is the coefficient for the one-sided

Table 2 . 11 :

 211 Comparison of fatigue limits of carbon steel

	Material	DC01	Carbon steel	C20 steel	MCS 1018 steel	C45 steel	SAE 1045 steel
	Ultimate stress (MPa)	430.9	-	520	520	778	710
	Fatigue limit (MPa)	280.26	268.7	265	263.2	274	300
	Load mode	Bending	Bending	Tension	Tension	Tension	Tension
	Load frequency	205	-	-	15	15	5
	Number of cycles	10 6	-	2 × 10 6	10 6	-	10 6
	Test method	Staircase test	-	Staircase test	S-N fitting	Staircase test	Intrinsic thermal dissipation
	Reference	This study	Papuga [89] Delahay et al. [90] Liakat et al. [83] Colombo et al. [91]	Teng et al. [92]

  N is the number of samplings 3 Resampling N data from X, to produce n times combination data X, which is donated as X = [x 1 , x 2 , ..., x N ]Estimate fatigue limit distribution by the same DM method 8 end foreach 9 Calculate the uncertainty factors.

4 foreach combination data X i do 5 Regard X i as a new staircase test results 6 Shuffle X i randomly 7

Table 3 . 1 :

 31 Effect of dimensions on the resonant frequency

	Specimen ID	Thickness (mm)	Length (mm)	Resonant frequency (Hz)
	S1	1	100	105
	S2	2	100	207
	S2L	2	170	107

Table 3 . 2 :

 32 Comparison of median and standard deviation (with DM)

	Specimen ID	Thickness (mm)	Length (mm)	Resonant frequency (Hz)	Median (m)	Standard deviation (s)
	S1	1	100	105	1417.51	41.41
	S2	2	100	207	1366.67	25.96
	S2L	2	170	107	1312.21	56.43

Table 4 . 1 :

 41 Overview of existing evaluation techniques for staircase testing

	Methods	Algorithms	Prior Distribution	Remarks
	Dixon-Mood method [27] (DM)			-Simplification -Recommended by ISO standard [35] -Strong dependence on step size
	Svensson-Lorén [38] correction on DM	MLE	Normal	-Function of sample size
	Braam-Zwaag [104]			-Function of sample size and step size
	correction on DM			-Worse estimation with small sample size
	Pollak's [38] correction on DM			-Function of sample size and step size
	Bootstrap [38, 98]			-Reduce the estimation uncertainty -Increase in the amount of computation
	MLE with censored data [40]	MLE	Yes	-Classical method
	Correction on			-Function of sample size
	MLE [39]			-Valid only for logarithmic standard deviation
				-Applicable to variable step size
	Zhang-Kececioglu [105] method (ZK)	Suspended item and MLE	Yes	-Worse than DM in the negative skew distribution [23] -Underestimate the standard deviation[23]
				-Overestimate evaluation on mean [24]
	Generalized MLE [24]	S-N curve and MLE	Yes	-Incorporate fatigue life in likelihood function -Paired failure-survival specimens
	Wallin's [42]	Binomial Probability(BP)	Use BP to select	-Binomial Probability to fit staircase data
	method	and MLE	distribution	-Strong dependence on step size
	IABG [40] method	MLE	Normal or Lognormal	-Based on logarithmic values of data -All specimens are evaluated -discarded invalid data, add fictitious data
	Deubelbeiss' [40] method	Probability of failure	No	-Fitting from probabilities of failure -Large bias in estimation

  NON-PARAMETRIC EVALUATION METHOD AND SIMULATION PROCEDUREEq. 4.10 and the improved bandwidth given in Eq. 4.12 are shown in Fig.4.2. It is obvious that the improved bandwidth leads to a smooth estimated PDF.

3.2, and α = 1.15 is determined in the end. Based on experimental data, the KDE estimation with Scott's rule, given in 4.2. Figure 4.2: Estimated PDF from different bandwidths

Table 4 . 3 :

 43 Parameters in simulation work

		Probability Distribution	CV	Evaluation method	Variable	Result
	Case 1	Normal	0.025
	Case 2 Case 3 Case 4	Normal Normal Lognormal	0.0125 0.05 0.025	KDE MLE DM	n = 10 to 100 d/s 0 = 0.1 to 2	(m k , s k ) for k = 1, • • • , 1000
	Case 5	Weibull	0.025

Table 4 .

 4 4: J-S divergence with different α parameter (×10 -2 )

	Case	Setup TRUE	1	1.05	1.1	1.15	1.2	1.25	1.3	1.35	1.4
	Normal CV=0.025	n=30 dn=1.0	1.75	2.08	1.97	1.88	1.81	1.76	1.72	1.71	1.69 1.51
	Normal CV=0.025	n=30 dn=0.1	51.04 49.51 49.09 48.67 48.25 47.82 47.39 46.96 46.53 46.10
	Normal CV=0.025	n=30 dn=2.0	9.83	9.33	9.44	9.59	9.76	9.94 10.14 10.34 10.55 10.76
	Normal CV=0.05	n=30 dn=1.0	0.88	1.59	1.33	1.14	0.98	0.86	0.76	0.68	0.61 0.56
	Normal CV=0.0125	n=30 dn=1.0	32.13 30.53 31.01 31.48 31.92 32.35 32.75 33.13 33.49 33.83
	Logormal CV=0.025	n=30 dn=1.0	1.91	2.18	2.08	2.00	1.94	1.90	1.88	1.87	1.88 1.90
	Logormal CV=0.025	n=30 dn=0.1	50.57 48.99 48.57 48.14 47.71 47.28 46.85 46.42 45.98 45.54
	Logormal CV=0.025	n=30 dn=2.0	9.84	9.35	9.46	9.61	9.78	9.96 10.16 10.36 10.57 10.79
	Logormal CV=0.05	n=30 dn=1.0	13.09 13.36 12.91 12.47 12.04 11.62 11.20 10.78 10.37 9.97
	Logormal CV=0.0125	n=30 dn=1.0	15.07 15.05 15.31 15.59 15.89 16.20 16.52 16.86 17.20 17.55
	Weibull CV=0.025	n=30 dn=1.0	18.07 18.64 18.34 18.05 17.78 17.52 17.26 17.02 16.79 16.56
	Weibull CV=0.025	n=30 dn=0.1	37.20 34.21 33.80 33.38 32.96 32.54 32.12 31.71 31.29 30.87
	Weibull CV=0.025	n=30 dn=2.0	15.77 16.72 16.45 16.24 16.08 15.95 15.84 15.77 15.71 15.67
	Weibull CV=0.05	n=30 dn=1.0	19.82 20.59 20.13 19.69 19.26 18.83 18.41 17.99 17.58 17.17
	Weibull CV=0.0125	n=30 dn=1.0	32.29 32.73 32.56 32.42 32.30 32.19 32.10 32.02 31.96 31.92
				dn: Normalised step size

Table 4 . 6 :

 46 The median and standard deviation from different estimation

	Methods	Median	Standard deviation
	DM	1366.67	25.96
	MLE (Normal distribution) 1366.71	22.51
	KDE (h = 11.56)	1366.66	22.61 12.21(corrected by Eq. 4.13)
	Combining the estimated distributions and Tab. 4.6, the following results can be

  .2.

	1			2
	( ) P θ Possible stress Init prior	Sa,1	prior Possible stress ( | ) P x θ 1	Sa,2
	Utility function	Test	Utility function	Test
	Sa,1 Best stress	Posterior 1 ( | ) P x θ	Sa,2 best stress	Posterior
	i-1			i
	prior		prior
	Possible stress	Sa,i-	Si	Possible stress	Sa,i
	Utility function		result Utility function	Test
	best stress		best stress Posterior	Posterior
	Sa,i-1		1 i ( | ) P x θ :	Sa,i

Table 5 . 1 :

 51 Different input parameters

		Probability Distribution Median range Standard deviation range
	Case 1	Normal	300-500	1-20
	Case 3	Normal	200-600	1-20
	Case 2	Normal	100-700	1-20
	Case 4	Normal	200-600	1-30
	Case 5	Normal	200-600	1-40

Table 6 . 1 :

 61 Different treatments for Ti-6Al-4V specimens used in Fig.6.1

	Curve	Vibration stress relief	Heat-treatment (℃)	Resonant frequency (Hz)	Strain (µm/m)
	C1-1200	-	-	331	1200
	C1-1500 C2-700 C2-900 C2-1200 C2-1500	-√ √ √ √	-----	330 359 353 353 357	1500 700 900 1200 1500
	C3-3000	-	650	340	3000
	C4-900	-	1050	335	900
	C4-1200	-	1050	327	1200

Table C . 1 :

 C1 Constants used in standard deviation correction (Eq. C.6) But the effectiveness of this correction is not good enough, for staircase tests with step size too large or too low may have errors magnified.

	Number of specimens (n)	A	b
	8	1.30 1.72
	10	1.08 1.10
	12	1.04 0.78
	15	0.97 0.55
	20	1.00 0.45
	30	1.00 0.22
	50	1.00 0.15
	PO standard deviation shows a more robust than Svensson-Lorén correction in
	different step size.

  E.3. WEIBULL DISTRIBUTIONConsequently, it may approximate the normal pdf, and for λ > 3.7 it is negatively skewed (left tail).As shown in Fig.E.2, The Weibull distribution with desired mean 400 and desired standard deviation 10 are created with 100000 data.Figure E.3: Weibull distribution testFor the observed data, µ X = 399.9766 and s X = 10.0523, m X = 401.5406.

	Annexe F
	Résumé en français	ÉVALUATION
	PROBABILISTE DES
	LIMITES DE FATIGUE F.1. Introduction générale
	ET AMÉLIORATION
		PAR DES ESSAIS
	EXPÉRIMENTAUX ET
		NUMÉRIQUES
		Lujie Shi
	Laboratoire de Mécanique de Normandie
	INSA Rouen Normandie, France

  'écart type et de la conception du protocole d'essai basée sur le théorème de Bayes.Chapter F.3 : description d'une méthode d'essai expérimentale de contrôle des vibrations basée sur la déformation réalisée sous agitateur électro-dynamique. Ce chapitre détaille une procédure d'essai en escalier avec un banc de flexion vibratoire et la sélection des paramètres. Les résultats des essais avec cette méthode offrent un ensemble de données pour la quantification des incertitudes et l'étude statistique.Chapter F.4 : présentation du Bootstrap utilisé pour caractériser la moyenne et l'écart type de la distribution de la limite de fatigue à partir des données issues de la méthode des escaliers, ainsi que l'incertitude associée à cette méthode d'essai. Les résultats comparent la distribution de la limite de fatigue du DC01.Chapter F.5 : traitement statistique des données d'essais de fatigue sur éprouvettes. Une nouvelle méthode d'évaluation basée sur l'estimation de la densité du noyau est proposée pour estimer la limite de fatigue à partir des essais en escalier. Une étude basée sur la simulation est réalisée en utilisant plusieurs distributions de probabilité avec différents coefficients de variation.

2 : court revue de la limite de fatigue et de la méthode de l'escalier démontrant le but du développement de la méthode d'essai expérimentale de contrôle de la déformation, de la technique d'évaluation des données avec cor-rection de lChapter F.6 : proposition d'un processus de conception de test avec le théorème bayésien en ce qui concerne à la fois la considération scientifique et l'exigence d'ingénierie.

(a) Response velocity at same strain level (b) Excitation acceleration in same strain level

(a) (b)

In this work, we have carried out the tests on the bending mode. Another way would be to perform the tests in torsion mode. The torsional fatigue limit could then be used to build a reliability model as presented in the work of Lambert et al.[START_REF] Lambert | A probabilistic model for the fatigue reliability of structures under random loadings with phase shift effects[END_REF].

Acknowledgments

Abbreviations TEST FOR IMPROVING FATIGUE LIMIT ASSESSMENT , corrected Scott's rule (SC), Silverman's rule (SI) and Sheather-Jones (SJ) method are illustrated by the red line, blue line and orange line, respectively. Each vertical bar's top and bottom represent the 5 th and 95 th percentile. The central marker is the estimated median from the 1000 staircase trials. From the comparison of the results obtained with the three bandwidths, it can be concluded that:

(1) From Fig. 4.4(a) and Fig. 4.5(a), three kinds of lines overlap, which means that the estimation of the median values is not sensitive to the bandwidth selector. (2) Fig. 4.4(b) shows that the bandwidth of all methods has no effect on the standard deviation when the number of specimens increases (higher than [START_REF] Cetin | The fatigue limit: An analytical solution to a Monte Carlo problem[END_REF][START_REF] Li | Evaluation of the Methods for Estimating the Fully Reversed Unnotched Fatigue Limits of Steels[END_REF][START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF][START_REF] Nicholas | Step loading for very high cycle fatigue: STEP LOADING FOR HCF[END_REF][START_REF] Lipski | Determination of Fatigue Limit by Locati Methodusing S-N Curve Determined by Means of Thermographic Method[END_REF][START_REF] Boresi | An appraisal of the prot method of fatgiue testing Part I[END_REF]. (3) According to Fig. [START_REF] Liu | Fatigue limit prediction of notched plates using the zero-point effective notch stress method[END_REF].5(b), the standard deviation estimation from these three bandwidths are all affected by the step size significantly. (4) Focus on Fig. 4.5(b), the difference in the three bandwidth kinds is reflected in the normalised d > 1. However, the relative difference of the three methods is smaller compared to the absolute estimation error.

The estimations from corrected Scott's rule (SC) are closed to the true values when the normalised step size is equal to 1. The mean J-S divergences of the three bandwidth selectors in the case of n = 30 are 0.02115 (for SC), 0.03231 (for SI) and 0.04982 (for SJ). It is clear that the SC gives the smallest J-S divergence and is therefore applied to the proposed KDE approach. However, Fig. 4.5(b) shows that the estimated standard deviation obtained from SC is still far from the true value. Therefore, a correction is introduced to improve the results of the standard deviation estimation.

Comparison of the probability distribution estimation

To complete the performance evaluation of the proposed method in terms of median and standard deviation estimation, the J-S divergence is introduced in this section as a means of evaluating the estimate quality with respect to the probability distribution. The average J-S divergence DJS reported in Tab. 4.5 is computed from 1000 trials in each case. From case 1 to case 3, all the investigated methods provide a stable estimate of the probability distribution, as the average J-S divergence is nearly constant for each method throughout these cases, which reveals that the estimate quality is less affected by CV. The DKDE JS is always smaller than that of the DM and MLE methods. This means that the KDE-estimated probability distribution is relatively less different from the true probability distribution.

As for the cases 1, 4 and 5, the DKDE JS is also better than that of the MLE method. However, both DKDE JS and DMLE JS are increased in case 5. Considering that the KDE method has offered a good estimation of the median and standard deviation in the same study case, the reason for the increase in DKDE JS is probably caused by the negative skewness of the Weibull distribution. Since the kernel function used by KDE in this work is symmetric, the estimated probability distribution is relatively less accurate when the underlying distribution is highly skewed. This leads to future research on the kernel functions used in the proposed KDE-based evaluation method.

In this study, the proposed KDE method is simulated and compared with different distribution models and different CV. Based on the simulation study, it is possible for the proposed method to generalize for many material. in Magazzeni et al. [START_REF] Magazzeni | Bayesian Optimised Collection Strategies for Fatigue Testing -Constant Life Testing[END_REF] study, this part introduces the Latin Hypercube Sampling (LHS) into the Bayesian framework. The same procedure is repeated identically for other samples with BMES until the posterior has a convergence. The convergence means only one distribution corresponds to the maximum posterior. In the next step, the LHS is applied to determine the stress amplitude and update the posterior.

Then, the LHS and BMES are alternately used to determine the stress amplitude. The posterior is updated after each test. Finally, the posterior P (θ|x 1:n ) is obtained after the total n specimens.

The resulted posterior, P (θ|x 1:n ), presents the probability of the observations under the given possible model parameters. The estimated model parameters are obtained by searching the maximum in the posterior after the last step (last specimen). In summary, the improved staircase protocol is detailed in Algorithm 5.1. The estimated median m and the estimated standard deviation s of the fatigue limit distribution Compared to the Bayesian staircase method in Section 5.3.3, this Bayes-LHS method enriched it in two areas: Firstly, the principle is kept since the stress amplitude must decrease after a failure result. In contrast, the load level must increase after a survival result. Secondly, the LHS is applied as a tool to select the stress amplitude, which offers more choice for stress amplitudes.

Example of Bayes-LHS procedure

Based on the Normal distribution assumption, an example in the case of n = 30 is presented in this section. The up-and-down figure from Bayes-LHS is shown in Fig. 5.8.

Similar to the conventional staircase procedure, the specimens are tested separately. The first few specimens perform a preliminary search of the median fatigue limit, and the test data gradually converge towards the solution. After posterior convergence to a single estimated distribution, the BMES and LHS alternate in picking Appendix A

Dynamics analysis of the specimen

Before the experiment fatigue test, it is desirable to obtain the dynamics characterization of the specimen by numerical model. In this study, the numerical model is established by finite element analysis (FEA) based on Ansys software 1 . It can be seen that the resonant frequency and the strain response from simulation and experiment are close. These results provide verification for the FE model .

A.1. Finite element modelling

A.2.3. Stress concentration factor

At the notch area of the specimen, the local stress is much greater than the nominal stress, which is known as stress concentration. Fatigue failure typically occurs at the notched edge because the local stress is much greater than the nominal stress. To simplify the description, two sample models are established: specimen without notch (SWON) for nominal stress, and specimen with notch (SWN) for maximum stress.

The Stress Concentration Factor (SCF) can be expressed as:

σ max is the local maximum stress and σ nom is nominal stress. Assuming the axial force (F = 1N ) and the dimension of specimen without notch, σ nom is given by:

This result is in accordance with the results of Ansys. Then, the static SCF is calculated as:

In terms of the fatigue limit, tests indicate [START_REF]Fatigue testing and analysis: theory and practice[END_REF] that the presence of a notch on specimen under cycling nominal stresses reduces the fatigue strength of the smooth specimen by a fatigue notch factor K t given by: K f = unnotched fatigue limit notched fatigue limit (A.4)

In general, K f is equal to or less than K t . The K f is related to both the notch root radius and ultimate tensile strength. More information about K f can be found in Liu et al. [START_REF] Liu | Fatigue limit prediction of notched plates using the zero-point effective notch stress method[END_REF]. In conclusion, the strain value at the edge and the middle of the specimen reduced section are almost the same as predicted by the numerical model in Fig. A.8. Considering the feasibility of sticking the strain gauge, the difference in the two locations is neglected, and the strain gauge is glued on the center of the specimen for all tests.

Appendix B

Staircase test data

The detailed data for the specimens in the staircase tests are listed in this section. The specimen names are listed in Tab. B.1. Appendix C

Comparison study using the Dixon-Mood method

C.1. Dixon-Mood method

The Dixon-Mood (DM) method was proposed by Dixon and Mood [START_REF] Dixon | A method for obtaining and analyzing sensitivity data[END_REF] in 1948, which provides approximate formulas of maximum likelihood estimation to calculate the mean and standard deviation of the fatigue limit. This method is a development and simplification from Maximun Likelihood Estimation (MLE) and the parameter distribution is assumed to be a normal distribution. Though there are many postprocessing methods that have been studied in recent years, DM is still classic and the most widely used method for staircase data.

In DM method, the less frequency event (survival or failure) is used to calculate the following quantity. The load levels S a are discrete by step size d and ordinally numbered by j, where j = 0 for the lowest load level S a,0 . Denoting by n c,j the number of the fewer frequency events (survival or failure) at the load level j, two auxiliary values A and B can be calculated by Eq. C.1:

The auxiliary values were used to estimate the mean m DM by Eq. C.2 and the standard deviation s DM by Eq. C.3. The minus sign given in Eq. C.2 is used if the failed specimens are evaluated and otherwise, the plus sign is used.

In which the subscript "DM" represents the Dixon-Mood method. The DM equations for the staircase method are also recommended by ISO 12107 [START_REF]Metallic materials-Fatigue testing -Statistical planning and analysis of data[END_REF].

C.2. Standard deviation correction for DM method

The small sample in the staircase test means that a lot of samples are needed under high confidence and high probability, but it is actually impossible to do so much. There is a drawback to the staircase test. The staircase method test is unbiased for the mean value estimation, and the standard deviation estimation is inherently low [START_REF] Carlos | Statistical Approaches Applied to Fatigue Test Data Analysis[END_REF][START_REF]D. Fuzes, ignition safety devices and other related components, environmental and performance tests for[END_REF]. One reason is that the staircase method focuses on the majority of the data points near the mean, it is more difficult to get an accurate measure of dispersion Randall D. Pollak [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF][START_REF] Randall D Pollak | Analysis of Methods for Determining High Cycle Fatigue Strength of a Material With Investigation of Ti-6Al-4V Gigacycle Fatigue Behavior[END_REF] gave detailed research discussing standard deviation estimation and presented a bootstrap correction method.

The deviation bias is a function of both step size(d) and sample size(n) in staircase testing [START_REF] Pollak | A simulation-based investigation of the staircase method for fatigue strength testing[END_REF]. The staircase method cannot eliminate or avoid this problem, but it can minimize the error of the standard deviation estimation value in terms of optimizing the experimental design [START_REF] Kim | Statistical uncertainty in the fatigue threshold staircase test method[END_REF], improving the data processing [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF] and creating correct coefficient.

It can only minimize the error of the variance estimation value in terms of optimizing the experimental design, improving the data processing, and modifying the coefficients. Brownlee [START_REF] Brownlee | The up-and-down method with small samples[END_REF] corrected the estimation of the mean to improve the situation that the first few trials all have the same outcome, and conclude that over 200 trials would be required to estimate standard deviation within 20% with the confidence of 95%. Some researchers put forward the corrections for DM method for better estimation of standard deviation.

Svensson-Lorén Correction (SL)

In Svensson's simulation works [START_REF] Svensson | Random Features of the Fatigue Limit[END_REF], a linear correction factor was proposed to improve maximum likelihood evaluation procedures.

C.3. Comparison results and discussions

Let's give two figures about median estimation with respect to the number of specimens and step size. From the above figures, the median value from DM method always approaches the true value and is almost independent of the number of specimens and the step size. Next, we will focus on the standard deviation estimation and its corrections (

C.4. Conclusions

It can draw the same conclusions from the figures. For both SL and PO corrections, the results with bootstrap enlarge the bias. if we only focus on SL corrections, the figures also give the same conclusion. Though bootstrap reduced the estimation scatter, it increased the bias and computational cost.

The SL correction improved the standard deviation estimation by a simple equation. Comparatively speaking, DM with Svensson-Lorén(SL) Correction gives the best estimation for standard deviation.

Appendix D

Comparison study with Maximum Likelihood Estimation

D.1. MLE method

The Maximun Likelihood Estimation (MLE) method is the most common technique to analyse the staircase data [START_REF] Roué | Simulation-based investigation of the reuse of unbroken specimens in a staircase procedure: Accuracy of the determination of fatigue properties[END_REF][START_REF] Bai | Measurement and estimation of probabilistic fatigue limits using Monte-Carlo simulations[END_REF] and general fatigue test data. The purpose is to use the known sample results to infer the most likely (maximum probability) parameter values that lead to such results. The MLE can be used to staircase data with variable step size and any kind of distribution. It is able to derive the estimated distribution parameters, however, the MLE needs a prior assumed distribution.

MLE method

The goal of the MLE is to combine the observations of each specimen in order to estimate the initial distribution. A likelihood function is created to evaluate the parameters of this distribution. In the general case with a dataset containing n samples, the likelihood function L is defined as:

where the f is the PDF of assumed distribution, which is the function of the distribution parameters, such as shape, loc, scale and so on. S a,i is the load amplitude of the specimen i. The evaluation of the distribution parameters is performed by maximizing function L.

MLE with censored data

Considering the data set X={ x 0 , ..., x j , ..., x n } resulting in the staircase approach,

x j being a state of survival or failure experienced by a specimen (without censor). When the fatigue strengths S a,j of the n specimens are known, For the staircase data, the specimen with failure and with survival gives different information of fatigue limit: the fatigue limit is lower than the applied stress (left censored) in case of failure and the fatigue limit is higher than the applied stress (right censored) in case of survival ESTIMATION (run-out). , if the specimen failed at the load level S a , the information we derive from the test is that S f < S a . It corresponds to left censoring and the observed probability is then described using the CDF as Eq. D.2:

While it can be known that survival can occur as a result of the load level is less than the corresponding fatigue limit, that is S f l > S a . The right censoring can be expressed as Eq. D.3:

where F is the CDF of the distribution. The maximum likelihood function L is expressed as Eq. D.4

The maximization of L results in the estimation of the distribution parameters. The logarithm of the minimized equation is strongly recommended in order to reduce some numerical computational problems. The new function to optimize minimum is:

Appendix E Distributions: Normal, Lognormal and Weibull

Note that the Normal, Lognormal and Weibull distribution are implied in several studies on staircase test, and represent non-skewed, negative skewed and positive skewed distributions respectively. Hence, this section describes the definition and properties of three distributions. Based on Scipy [START_REF] Virtanen | SciPy 1.0: fundamental algorithms for scientific computing in Python[END_REF] statistics package (scipy.stats), the formulas are uniformly expressed with location, scale and shape parameters.

For each distribution, the definition of Probability Density Function (PDF) and Cumulative Density Function (CDF) are given. Then we introduce the statistical properties and transformation of distribution parameters and population parameters. Lastly, an example of the distribution with desired mean 400 and desired standard deviation 10 are created. Total 100000 data are sampled from the distribution. And we can obtain the mean, standard deviation and median of sampled data (population), which are consistent with desired mean and desired standard deviation.

In the following formulas:

E.1. Normal distribution

The PDF and CDF of Normal distribution is given such as:

where α, β are location (mean) and scale (standard deviation) parameters respectively. The statistical parameters of Normal distribution can be obtained:

The distribution with desired mean 400 and desired standard deviation 10 are created, and 100000 data are sampled from the distribution. a sample is illustrated in