5 Identification and validation of the incremental model

1.27 Crack opening strain in a variable amplitude test [START_REF] Vormwald | A fracture mechanics based model for cumulative damage assessment as part of fatigue life prediction[END_REF] .......... 1.28 Multi-scale approach of condensed model ................. 1.29 Strain-stress curves under tensile load at three temperatures for 316L SS . 1.30 Traction-compression behavior A: stress-strain curve with Bauschinger effect. B: with isotropic hardening. ..................... 1.31 Cyclic hardening and softening of 316L SS in strain-controlled cyclic tests 1. [START_REF] Elber | The significance of fatigue crack closure[END_REF] Ratcheting strain for experiments performed on 316L (a) at room temperature with increasing mean stress and constant stress amplitude [START_REF] Chaboche | On some modifications of kinematic hardening to improve the description of ratchetting effects[END_REF]. (b) at 300K with increasing maximum stress and constant mean stress [START_REF] Feaugas | Ratchetting process in the stainless steel aisi 316l at 300 k: an experimental investigation[END_REF] . 1.33 (a). Hardening stages for 316L SS subjected to LCF under different temperatures (b). Stress amplitude evolution for 316L SS subjected to LCF under different temperatures for ∆ε p = ±2.10 -3 [5] ............ 1.34 Evolution of the fatigue life with temperatures for three plastic strain amplitude [5] observation. An incremental crack propagation model proposed by S. Pommier et.al [START_REF] Pommier | Cyclic plasticity of a cracked structure submitted to mixed mode loading[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF] is adopted in this study. The extension of this model from SSY conditions to LSY conditions is necessary so as to predict the LSY fatigue crack growth rate.

Thesis organization

The thesis is composed of six chapters. The first chapter is the state of art and the literature review about the crack propagation. A review of the general knowledge of fracture mechanics and the fatigue crack propagation models available from small scale yielding to large scale yielding is presented. A brief introduction of an incremental model [START_REF] Pommier | Cyclic plasticity of a cracked structure submitted to mixed mode loading[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF] of fatigue crack growth modelling is given. A literature study of 316L stainless steel particularly of its cyclic behavior is presented as well.

The second chapter is devoted to the introduction of the experimental set-ups. A PACIFIC set-up is developed by EDF in the exploration of crack growth in large-scale yielding conditions under cyclic thermal loadings. However, limited by the extraordinary cost of a PACIFIC test (time and sum of money), only one PACIFIC test can be carried out in the study. Therefore, the objective of this chapter is the proposition of a simplified setup including the test specimen geometry in the representative of PACIFIC set-up which allows conducting LSY fatigue crack propagation tests under mechanical loadings so as to provide and enrich the experimental data about LSY fatigue crack. To do so, an elasticplastic cyclic constitutive law for 316L stainless steel is required and identified for a large strain range.

The third chapter is dedicated to the study of fatigue crack growth in large scale yielding condition in 316L SS. The tests are performed on the simplified mechanical set-up as well as the PACIFIC set-up. Experimental results obtained from both the tests are exploited with macro and micro approaches. This chapter gives an insight in the mechanism of LSY fatigue crack growth in this material, which constitutes the fundamental theory for the crack propagation modelling.

The fourth chapter deals with the development of the incremental model proposed by S. Pommier et al. for mode I crack propagation [START_REF] Pommier | Cyclic plasticity of a cracked structure submitted to mixed mode loading[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF] from small scale yielding conditions to large scale yielding conditions. Influence of the large-scale plasticity on the crack propagation law and the plastic blunting law of the incremental model are analyzed respectively by numerical simulations. Material effect in consideration of different hardening behaviors is examined as well. An incremental model for mode I crack growth in LSY conditions for materials exhibiting combined isotropic and kinematic hardening is determined at the end of this chapter.

Once the incremental model for mode I fatigue crack growth in large scale yielding conditions is determined, the fifth chapter is dedicated to the identification and the confrontation of numerical results given by the incremental model with experimental results. The identification process of the parameters in the incremental model with both the numerical simulations and the experimental results of mechanical tests is presented. The interpretation of PACIFIC test with the identified incremental model is performed in the end. known a priori.

The stress and displacement fields are for a crack in an infinite plane are presented here. The solutions are given on strain plane for mode I and II.

8 > > > > > > > < > > > > > > > : 3θ
2 )

σ I zz = ν(σ I xx + σ I yy ) 8 > > > > > > > < > > > > > > > : 3θ
2 )

⌘ σ I zz = ν(σ I xx + σ I yy ) 8 > > < > > : u I x = (1 + ν)K I E r r 2π cos( θ 2 ) 3 -4ν -cos(θ) u I y = (1 + ν)K I E r r 2π sin( θ 2 ) 3 -4ν -cos(θ) 8 > > < > > : u II x = (1 + ν)K II E r r 2π sin( θ 2 ) 3 -4ν -cos(θ) u II y = - (1 + ν)K II E r r 2π cos( θ 2 ) 3 -4ν -cos(θ)
The stress intensity factor is not only used as a parameter defining the amplitude of crack tip singularity, but also used as a parameter describing the onset of crack growth or fracture. The critical value in Mode I is termed as "fracture toughness" and labelled as K IC . For fatigue problems, ∆K is used, where ∆K = K max -K min of a cycle.

However, most industrial materials exhibit an elastic-plastic behavior. In these cases, the stress intensity factor defined linear elastic crack tip field is valid only when the plasticity is confined to a small region of the crack tip, as illustrated in Fig. 1.7, where the r min is much smaller than r max . While the crack tip undergoes significant plasticity, the size of plastic zone engulfs the K dominance region, K no longer characterizes stresses near the crack tip. Then two alternative parameters are proposed to describe the crack tip condition in elastic-plastic material, namely J-integral and CTOD (crack tip opening displacement). Each of the parameters could be used as a fracture criterion for relatively large amount of crack-tip plasticity.

ε ε 0 = σ σ 0 + α( σ σ 0 ) n (1.3)
where σ 0 refers to the reference stress value which usually equals to the yielding strength. ε 0 = σ 0 /E. α is a dimensionless constant and n corresponds to strain-hardening exponent.

σ ij = k 1 ( J r ) 1 n+1
(1.4)

ε ij = k 2 ( J r ) n n+1 (1.5)
where k 1 and k 2 are proportionality constants. When n=1, these equations are consistent with LEFM theory with a 1/ √ r singularity. Hence it is observed that the J could completely describe the conditions within the plastic zone.

However, the same anomaly as in LEFM appears to HRR solution, namely both predict infinite stresses as r ! 0. This analysis leads to the breakdown of HRR solution in the crack tip region where large strain and crack blunting take place. Therefore, the J integral could be considered as fracture criterion as long as there is a region surrounding the crack tip whose condition could be describe by the HRR solution.

The application of J-integral in fatigue situation become complicated due to its definition. As J-integral is only path-independent for linear or non-linear material for a monotonic loading, so when unloading occurs in an elastic-plastic material, the J-integral is no longer valid. Therefore, the application of J-integral has always been in question. In spite of these argument, some researchers have successfully used ∆J under certain conditions to correlate the fatigue crack growth with a definition of ∆J as follows [START_REF] Dowling | Fatigue crack growth during gross plasticity and the j-integral[END_REF][START_REF] Lamba | The j-integral applied to cyclic loading[END_REF][START_REF] Wüthrich | The extension of the j-integral concept to fatigue cracks[END_REF]:

The material ahead of a fatigue crack experiences cyclic elastic-plastic loading. The material's deformation can be characterized by the stress range ∆σ ij and the strain range ∆ε ij in a given cycle. Fig. 1.9 shows a cyclic stress-strain curve where the stress and strain increase from initial values of σ 1 ij and ε 1 ij to σ 2 ij and ε 2 ij under the loading branch. Then the ∆J could be defined: The size of the plastic zone at the crack tip increases with the applied loads and the elastic singularity is no longer valid. As long as the strain/stress field of crack tip region could be solved by HRR solution, the corresponding loading condition is considered as intermediate scale yielding(ISY) as illustrated in Fig. 1.15:b. In ISY, the Jintegral [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF] and CT OD [START_REF] Wells | The condition of fast fracture in aluminum alloys with particular reference to comet failures[END_REF] can be used as the crack-tip characterizing parameters.

As the applied loads increase, a large scale yielding (LSY) condition is obtained. However, there is no unique definition for the LSY condition. Some define the LSY condition as the situation where crack tip plastic zone size reaches the characteristic dimension of the cracked body (the crack length or the un-cracked ligament for example), as for the short crack cases where the crack tip plastic zone size is comparable to the crack length [START_REF] Dowling | Fatigue crack growth during gross plasticity and the j-integral[END_REF]. Others defines the LSY condition as the situation where the strain in crack tip region is so great that HRR solution breaks down in its description as illustrated in Fig. 1.15: c. Consequently, the crack tip could no longer be characterized by Jintegral or CT OD [7].

In both cases, it could be concluded that a large crack tip plastic zone appears in LSY condition.

In our study, the LSY condition is defined in the cases where plastic deformation occurs without presence of crack in a structure. As to fatigue crack in LSY condition, it signifies that cyclic plastic deformation takes place in a non-cracked structure under cyclic loading. This definition is in coherence with other definitions as with presence of a crack, the crack tip plastic zone is supposed to still remain important.

Mechanism of fatigue crack growth

Various models of fatigue crack propagation in ductile materials have been proposed from the earliest void linkage model proposed by Forsyth and Ryder in 1961 to Laird and Smith's plastic blunting model in 1962 as well as Neumann's coarse slip model introduced in 1969. However, none of these models has been universally accepted for that any of these model could fully explain the range of cracking mechanisms in different materials and over large range of stress levels encountered. Nevertheless, these models also provide useful insights into the types of processes occurring at the crack tips during fatigue crack propagation. Here, two models will be highlighted, one is Laird and Smith's plastic blunting model which is based on the crack tip blunting and resharpening mechanism [START_REF] Laird | The influence of metallurgical structure on the mechanisme of fatigue crack propagation[END_REF], the other is Neumann's coarse slip model [START_REF] Neumann | Coarse slip model of fatigue[END_REF].

Under small scale yielding condition

Plastic blunting model This model is based on the repetitive blunting and sharpening of the crack tip process due to plastic flow. Fig. 1.16 illustrates the mechanism of fatigue crack growth of this model. It can be seen that through the application of tensile load, highly localized deformation takes place along the slips planes of maximum shear stress oriented at 70 • to the crack plane (on plane strain). Upon further increase of load, the deviates from the linear trend at high and low ∆K levels (Region I and III). Paris and Erdogan [START_REF] Paris | A critical analysis of crack propagation laws[END_REF] were the first to propose the power-law relationship for fatigue crack growth in Region II with an exponent of 4. The proposed power law then became widely known as Paris' Law (Eq. 1.21), where C and m are material constants and could be determined experimentally.

da dN = C∆K m (1.19)
As Paris' Law is limited to describe the fatigue behavior in region II. A number of researchers have developed equations which allow modelling all or part of the da/dN vs. ∆K curve [START_REF] Foreman | Numerical analysis of crack propagation in cyclic-loaded structures[END_REF][START_REF] Klesnil | Influence of strength and stress history on growth and stabilisation of fratigue crack[END_REF][START_REF] Forman | Behavior of surface and corner crack subjected to tensile and bending loads in ti-6al-4v alloy[END_REF]. Forman and Mettu [START_REF] Forman | Behavior of surface and corner crack subjected to tensile and bending loads in ti-6al-4v alloy[END_REF] was the first to publish an expression to describe fatigue crack growth in all the three regions. The equation is given by Eq.1.20 where C,m, p, q are material constants. This equation is based on the hypothesis that region III behavior is caused by a superposition of fracture and fatigue rather than plastic zone effects. The R ratio dependence is also taken into consideration with

K max = ∆K 1-R . da dN = C∆K m (1 - ∆K th ∆K ) p (1 - K max K c ) q (1.20)
From these equations, a fatigue crack growth rate could be computed for a given material. The growth rate depends on loading parameters ∆K and K max . However, all these expressions assume elastic conditions and they are strictly limited in the prediction of fatigue crack rate under constant-amplitude loading without occurrence of crack closure; Therefore, for most fatigue cases under variable-amplitude load where history effect and crack closure phenomenon are observed to have a great influence on crack growth rate, these models are no longer available.

History effect

In practice, industrial structures usually experience a spectrum of loads over its lifetime which is far from constant-amplitude loads required by Paris' law. In such cases, with consideration of elastic-plastic material behavior, the fatigue crack growth rate at any moment depend not only on the current loading conditions but also on its history loading condition. This phenomenon is called history effect.

Retardation effect of an overload is a typical example of history effect for fatigue crack growth. Consider the fatigue loading illustrated in Fig. 1.20 on the left, constant amplitude loading is interrupted by a single overload. Such a fatigue behavior related to this overload could be observed: an acceleration in da/dN is seen in a brief period before a great da/dN deceleration, named Retardation is found. As the crack grows, the crack growth rate eventually approaches the growth rate under constant-amplitude loading prior to the overload.

Crack growth model under variable-amplitude loading

Several mechanisms have been proposed to explain the retardation effect following an overload, in which the plasticity-induced closure mechanism is most favorable. If crack closure occurs during constant-amplitude loading, the subsequent crack tip blunting causes the crack faces to move apart when overload is applied. Consequently, crack closure does not occur during the overload cycle even in the cycles following the overload, resulting in an immediate acceleration in fatigue crack growth rate. Once the crack grows a short distance into the plastic zone created by the overload, the compressive residual stress in the plastic wake make the crack faces to close and the effective stress intensity factor is therefore reduced, which as a consequence, results in a retardation of fatigue crack growth rate.

A number of models have been proposed to account for the overload effects in fatigue crack propagation. Wheeler [START_REF] Wheeler | Spectrum loading and crack growth[END_REF] and Willenborg [START_REF] Willenborg | A crack growth retardation model using an effective stress concept[END_REF] have proposed the retardation models based on the assumption that the compressive residual stress in front of crack tip influences the crack growth rate while Newman has developed a numerical model for plasticity-induced closure based on strip yield concept [START_REF] Newman | Prediction of fatigue crack growth under variable amplitude and spectrum loading using a closure model[END_REF].

Wheeler introduces a retardation factor φ to correlate crack growth rate given by Paris' law in order to consider the overload:

da dN retard = φ da dN non-retard (1.22)
where φ is determined by Eq.1.23:

φ =( r y(c) + ∆a r y(o) ) γ (1.23)
where γ is a fitting parameter, r y(c) refers to the plastic zone size of current amplitude and r y(o) corresponds to the plastic zone size of overload. The plastic zone size is estimated by Irwin's approach with r y(c) = 1 2π ( Kmax σ YS ) 2 and r y(o) = 1 2π ( Ko σ YS ) 2 on plane stress (Fig. 1.21).

Wheeler assumed that the retardation effects persists as long as the current plastic zone is included within the overload plastic zone and the effects disappears once the current plastic zone touches the outer boundary of the overload zone. This model gives a quite correct description of a single overload effect. However, when repetitive overloads or blocks of overloads are applied, the accuracy of this model is questioned. In fact, in tests with repetitive overloads in material with non-linear hardening, the retardation effect could be far too complex to be predicted by this model.

Unlike the Wheeler model, Willenborg has defined a residual stress intensity factor K R (Eq.1.24) which decreases with crack propagation in the overload plastic zone. An effective loading ratio R ef f is then introduced as in Eq.1.25. In order to predict crack growth rate with Willenborg's assumption, a crack growth expression with R dependence is used as shown in Eq.1.26 [START_REF] Walker | The effect of stress ratio during crack propagation and fatigue for 2024t3 and 7075-t6 alu[END_REF].

Up to now, a brief overview of fatigue crack growth models based on elastic/elasticplastic fracture mechanics is presented, from Paris' Law for elastic materials to ∆J or ∆CT OD based models for elastic-plastic materials in consideration of either constantamplitude or variable-amplitude loadings. According to loading conditions and with appropriate assumptions, some of these models could give quite correct fatigue crack growth prediction. However, because of the complexity of industrial material's behavior and the variety of industrial components' loading condition, application of these models could be limited to certain extent. The disadvantage of these models will be stated in the following section and an alternative model which focuses on time increment crack growth will be presented.

5 Incremental model [START_REF] Pommier | Cyclic plasticity of a cracked structure submitted to mixed mode loading[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF] Generally in laboratory conditions, fatigue crack propagation rate in metal is realized by Paris' Law [START_REF] Paris | A rational analytic theory of fatigue[END_REF] provided that the cracked structure is under constant-amplitude cyclic load and the material behaves elastically. However, the application of Paris' law is quite limited for industrial problems for a few reasons. Firstly, for most industrial component, the applied load in operating condition is more than constant-amplitude cyclic; Secondly, the material of the component exhibits an elastic-plastic behavior, which originates the history effects in fatigue crack growth, whose importance has been demonstrated and explained by [6,[START_REF] Dugdale | Yielding in steel sheets containing slits[END_REF][START_REF] Elber | The significance of fatigue crack closure[END_REF][START_REF] Elber | Fatigue crack closure under cyclic tension[END_REF][START_REF] Fleck | Influence of stress state on crack growth retardation[END_REF][START_REF] Forman | Numerical analysis of crack propagation in cyclic-loaded structures[END_REF][START_REF] Fuehring | Dugdale crack closure analysis of fatigue cracks under constant amplitude loading[END_REF][START_REF] Hamam | Variable amplitude fatigue crack growth, experimental results and modeling[END_REF][START_REF] Lemaitre | Mécanique des matériaux solides[END_REF][START_REF] Mcclung | Crack closure and plastic zone sizes in fatigue[END_REF][START_REF] Mcevily | Crack opening displacement and rate of fatigue crack growth[END_REF][START_REF] Newman | Mechanics of fatigue crack closure[END_REF][START_REF] Pommier | Cyclic plasticity of a cracked structure submitted to mixed mode loading[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF][START_REF] Pommier | Time-derivative equations for mode i fatigue crack growth in metals[END_REF][START_REF] Pommier | Time-derivative equations for fatigue crack growth in metals[END_REF][START_REF] Schijve | An evaluation of a fatigue crack growth prediction model for variableamplitude loading (preffas)[END_REF][START_REF] Schijve | Some formulas for the crack opening stress level[END_REF][START_REF] Wheeler | Spectrum loading and crack growth[END_REF][START_REF] Willenborg | A crack growth retardation model using an effective stress concept[END_REF]. The prediction of crack propagation rate under variable-amplitude cyclic load in consideration of history effect is out of reach for Paris' law.

If the material exhibits non-linear hardening, the loading path (not only the peak loads), has to be considered in the prediction of fatigue crack growth rate. Various crack propagation models (NASGRO, PREFFAS, Strip Yield...) have been proposed. However since most of the models allow predicting a fatigue crack growth rate per cycle, it requires the use of a cycle counting method (e.g. rainflow) based on load spectra recorded on the structures in operating conditions which might be quite far from cyclic. When the load sequence stemming from the cycle reconstruction differs significantly from the original one, the life prediction may be quite questionable.

In this context, an incremental propagation model has been proposed and developed by S. Pommier and al. at LMT [START_REF] Decreuse | History effect in fatigue crack growth under mixed-mode loading conditions[END_REF][START_REF] Doquet | Fatigue crack growth under non-proportional mixedmode loading in ferritic-pearlitic steel[END_REF][START_REF] Fremy | Load path effet on fatigue crack propagation in i+ii+iii mixed mode conditions -part 2 : Finite element analyses[END_REF][START_REF] Fremy | Load path effect on fatigue crack propagation in i + ii + iii mixed mode conditions -part 1: Experimental investigations[END_REF][START_REF] Hamam | Mode i fatigue crack growth under biaxial loading[END_REF][START_REF] Hamam | Variable amplitude fatigue crack growth, experimental results and modeling[END_REF][START_REF] Lopez-Crespo | Numerical analysis of crack tip plasticity and history effects under mixed mode conditions[END_REF][START_REF] Pommier | Cyclic plasticity of a cracked structure submitted to mixed mode loading[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF][START_REF] Pommier | A multi-scale approach to condense the cyclic elastic plastic behaviour of the crack tip region into an extended constitutive model[END_REF][START_REF] Pommier | Time-derivative equations for mode i fatigue crack growth in metals[END_REF][START_REF] Pommier | Time-derivative equations for fatigue crack growth in metals[END_REF][START_REF] Ruiz-Sabarieg | Oxidation assisted fatigue crack growth under complex non-isothermal loading conditions in a nickel base superalloy[END_REF][START_REF] Thieulot-Laure | A multiaxial fatigue failure criterion considering the effects of the defects[END_REF]. This model avoids any cycle reconstruction method by using an incremental crack propagation method in order to predict crack growth rate per second in complex loading conditions and in non-linear materials. This approach is based on the assumption that "pure" fatigue crack growth stems from crack tip plasticity [START_REF] Li | Vector ctd criterion applied to mixed mode fatigue crack-growth[END_REF][START_REF] Neumann | Coarse slip model of fatigue[END_REF][START_REF] Neumann | The geometry of slip processes at a propagating fatigue crack-ii[END_REF][START_REF] Pelloux | Mechanisms of formation of ductile fatigue striation[END_REF]. With such an assumption, an incremental model for "pure" fatigue crack growth could be derived from an incremental plasticity model for the crack tip region. The crack growth rate per second can be predicted from the measure of crack tip plasticity. The simplified model that has been developed at LMT aims at condensing all the effects of the non-linear behavior of the material in the crack tip region in a set of constitutive equations based on the minimum number of variables necessary to reasonably represent the problem of crack tip plasticity. A brief methodology of this model is presented in the following while more details of this model will be addressed in later chapter.

Methodology

This method is based on a multi-scale approach of a cracking problem in non-linear materials. The following hypothesis have been considered:

• Crack propagation during a loading increment is negligible compared to the dimension of plastic zone at crack tip. Thus it is unnecessary to take into consideration of crack front movement during one loading increment to determine plasticity increment in the crack tip plastic zone.

• For crack growth in small scale yielding conditions, the bulk of the cracked structure remains in elasticity. Hence the behavior of material in plastic zone at crack tip is strictly restricted by the stress of the non-plastic part. As a result, the number of degree of freedom in the description of the material behavior of the crack tip region is quite limited.

• Infinitesimal strain conditions are considered. Under such conditions, self-similarity of crack geometry is obtained by conserving of its symmetric plan in elastic-plastic conditions and with respect to change of scale. Therefore it is possible to represent the movement by a superposition of terms, each term corresponds to a product of an intensity factor and a spatial distribution which is invariant under change of scale.

• As it is always possible to obtain a transient elastic behavior by reversing the loading direction during a time increment despite the plasticity generated in previous increment, the elastic and plastic movements could be considered kinematically independent and each of the movement respects the hypothesis before.

Therefore, the movement could be partitioned into elastic and plastic parts with the least degree of freedom for each part for mode I crack growth problem. Considering the self-similarity, each part could be expressed as a product of a spatial distribution which is known a priori and an intensity factor which is used as kinematic degree of freedom.

Fig. 1.28 illustrates the methodology of the incremental and multi-scale method in a simplified manner. The first step is to identify a constitutive model for the studied material based on the cyclic elastic-plastic experimental results. The second step consists in EF modelling of a cracked structure with the identified material model in order to verify the necessary numbers of degree of freedom to represent the material's movement at crack tip region as well as to determine the spatial distribution (named also as reference field) for each degree of freedom. Once the reference fields are known, the post-treatment of EF calculations with more complex loadings will be performed so as to establish a series of curves describing the evolution of nominal stress intensity factor in relation with

316L stainless steel

The austenitic stainless steel 316L (316L SS) is largely employed as structural material in sectors such as aeronautics, pressure vessels and nuclear power plant not only because of its excellent mechanical properties but also for its ductility and high corrosion resistance in a humid environment with high temperature. In operating situations, these components could undergo complex conditions such as transients, instabilities, turbulence or other thermal stratifications, therefore, it is of interest to master the evolution of the material's mechanical properties in response to different loading conditions. Thermal-mechanical cyclic loadings are applied to this material in our study. Thus a bibliographic study focusing on 316L's mechanical properties under cyclic loading condition and different temperatures will be addressed in this part.

Chemical content

The austenitic stainless steel of Fe-Cr-Ni type is characterized by its low carbon content (less than 0.03%) and its high content in chromium (17-20%), nickel (9-14%) and molybdenum (2-4.5%). The full chemical content is shown in Table .2.1. The alloying elements and the low carbon content in the material not only improve its resistance to corrosion at elevated temperature but also increase its ductility and fracture toughness.

Mechanical properties under tensile load

The austenitic stainless steel exhibits a face-centered cubic crystalline structure. It has an initial yield stress (R p0.2% ) less than 300MPa, a maximum stress (R m ) below 650MPa and an elongation to fracture as high as 60% at room temperature. These mechanical properties could be improved by different hardening procedures. Temperature plays also an important role in its mechanical behavior under tensile loading particularly on the yield stress which is observed to decrease with the temperature as shown in Fig. 1.29 [START_REF] Le-Roux | Influence des paramètres métallurgiques et d'essais sur l'amorçage des fissusres de fatigue en déformation imposée des aciers inoxydables austénitiques[END_REF].

Mechanical properties under cyclic load 6.3.1 Bauschinger effect

The behavior of 316L stainless steel under cyclic loadings is primarily characterized by Bauschinger effect firstly observed in 1881 [START_REF] Bauschinger | Ueber die veranderung der elasticitatsgrenze und elastcitatsmodul verschiedener[END_REF]. It refers to a decrease in yield strength after a first plastic deformation of tensile load as shown in Fig. 1.30. plasticity. Instead of giving crack growth par cycle, it gives crack growth par increment. This approach allows predicting fatigue crack growth under complex variable-amplitude loading conditions for material's exhibiting elastic-plastic behavior.

An overview is also realized on the studied material-316L stainless steel. Material behavior under monotonic as well as uni-axial cyclic loading is investigated. Cyclic hardening/softening, cyclic ratcheting, load history effect are found for material's characterization. Temperature is also found to have an influence on material's behavior. This chapter aims at studying the experimental set-ups for fatigue crack propagation under LSY conditions. The available PACIFIC set-up is supposed to perform LSY fatigue experiment under certain thermal-mechanical loadings. However, its long cyclic duration makes PACIFIC test time-consuming and its quasi-structrual specimen makes the test quite expensive. Therefore, a simplified set-up with appropriate specimen is required in order to study the mechanism of LSY fatigue crack growth. The simplified set-up is required to be capable of performing stable LSY fatigue test under cyclic mechanical loadings. A comparable mechanical field to that observed on PACIFIC specimen should also be reproduced by the mechanical specimen.

Chapter 2 Material and experimental set-ups

The design of the simplified set-up for LSY crack propagation study is carried out with the numerical simulation, which needs a cyclic elastic-plastic constitutive law for 316L SS valid for significant strain levels. Therefore, the first section of this chapter is dedicated to the identification of a constitutive law for 316L SS based on low cycle fatigue (LCF) experimental data. The second section is devoted to the introduction and numerical study of the PACIFIC set-up. The last section deals with the design of the simplified mechanical set-up based on the numerical analysis of PACIFIC set-up.

Material-316L stainless steel

Cyclic behavior under important strain levels and under different temperatures is highlighted in this section. Low cycle fatigue tests have been carried out in the laboratory of EDF and a cyclic elastic-plastic constitutive law for 316L SS is identified based on the experimental data.

General properties of the studied 316L SS 1.Chemical content and microstructure

The studied material (316L SS or Z2CND17-12) is an austenitic stainless steel provided by Creusot-Loire-Industrie. It is sampled from a laminated plate labelled T217 in EDF. A chemical analysis has been conducted at MMC department of EDF [4]. Results are given in Tab. 2.1. It is observed that for the studied plate of 316L SS, there is no important difference of grain size in both rolling and transverse directions. The grains in interior part of the plate are slightly smaller than those close to the surface of the plate with a grain size of 80-120 µm in interior against 120-180 µm in exterior [4].

Thermal-elastic characteristics

The thermal properties of this material are given in Tab.2.2. Values of thermal expansion coefficients α are the mean values calculated between 20 • C and the considered temperature. Mechanical characteristics of this material are given in Tab.2.3. 1.2 Identification of cyclic mechanical behavior of 316L SS

T( • C) α (10 -6 m/m/ • C) λ (W /m/ • C) p λρC p (kJ/m 2 / • C/s 0.5 ) λ/(ρC p ) (10 -6 m 2 s -1 ) 20

Protocol of low cycle fatigue tests

The experiments were performed with the help of a hydraulic machine with a capacity of 100kN on a cylindrical specimen (Fig. 2.1). The specimen has a cylindrical useful zone with a length of 16mm and a diameter of 8mm. The tensile axis of the specimen is aligned with the rolling direction of the sheet. The test is strain-controlled using an extensometer which has a measurement range of 10mm and a measurement scale up to ±10%. The tests are conducted at different temperatures: 20 • C, 150 • C and 300 • C. A constant strain rate of 10 -3 s -1 and a loading ratio R ε of -1 are applied for all the tests.

model was adopted for this study. The parameters of this model were identified based on the experimental data presented in 1.2.2 by using an iterative approach in Code_Aster.

The identification is performed over a large range of strain amplitudes from 0.3% to 2.0%. The identified parameters for constitutive behavior of 316L SS for different temperatures are presented in Tab.1.1. It is to note here the parameter b of isotropic hardening is chosen in order to reach the material stabilisation as soon as possible, thus the evolution of maximum stresses with number of cycles can not be evaluated by this value. Comparison of stabilized experimental cycles and numerical cycles for different temperatures are plotted in Fig. 2.8-Fig. 2.9. In general, a good agreement is obtained. The constitutive law is able to produce the non-linear hardening and the same stress level for a large scale strain level. Meanwhile, this large scale identification cannot perfectly describe the cyclic behavior of all strain levels. As illustrated in Fig. 2.8, the stress is overestimated for the smallest strain amplitude whereas it is underestimated for largest strain amplitude at the early stage of hardening. A non-symmetry between the loading and unloading strain-strain experimental curves are observed for the last loading amplitudes, implying the presence of material damage cumulation.

The history effects observed in the experiments can also be simulated. Fig. 2.10 shows the history effect simulated by the constitutive law at 150 • C. It could be observed that the stress range of a given strain amplitude is larger than that for the same strain amplitude after the maximum loading, implying a load history influence to material behavior. Moreover, the smaller the strain level is, the larger the influence of history effect on the stress level could be. This is consistent with the experimental results.

η 1 1 1 Q 0 (MPa) 620 270 250 Q m (MPa)
-100 -50 -50

Conclusion

The studied material 316L austenitic stainless steel, has been presented in this section. More attention has been paid to the identification of a cyclic constitutive law for this ditions. For an isothermal test with cyclic thermal load varying from 40

Simulation of PACIFIC experiment

As discussed above, various thermal loading conditions can be applied during a PACIFIC test. In this section, we try to figure out a thermal loading which is capable of creating large scale yielding conditions in the mock-up of PACIFIC disc and thus allows to carry out Mode I fatigue crack propagation in LSY conditions. Moreover, the thermal loadings should also be as simple as possible to limit the test duration. An isothermal condition is proposed. Elastic and elastic-plastic numerical simulations of the test under this condition are performed to validate this assumption.

Definition of the loading conditions

The mechanism of thermal fatigue crack propagation in PACIFIC disc is as follows: cyclic thermal loadings result in cyclic thermal expansion of the material. If different temperatures are applied to different parts of the PACIFIC disc, the thermal gradient is produced which in return, results in heterogeneous thermal expansion, and hence results in mechanical stresses and strains in the PACIFIC disc. Due to the cyclic thermal loadings, the stresses vary cyclically, making the fatigue crack growth. Inspired by this mechanism, an isothermal loading condition is proposed. Cyclic thermal loading is applied to the complementary ring while the temperature applied to the mock-up remains constant and equals to the minimum temperature of the cyclic thermal loading. Thus a cyclic thermal gradient in radial direction is created, resulting in more thermal expansion in the complementary ring than in the mock-up. The deformed complementary ring causes the mock-up to deform as well. Therefore, a cyclic angular stress field is created in mock-up, which makes Mode I fatigue crack growth to happen.

With respect to the capacity of PACIFIC set-up, such a thermal load is defined: a cyclic thermal loading varying from 40 • C to 220 • C is applied on the top and bottom surfaces of the complementary ring while a constant temperature of 40 • C is imposed to the top and bottom surfaces of the mock-up as shown in Fig. 2.19. Compressive load of 23.8 MPa provided by hydraulic jack is applied to the complementary ring in order to maintain the disc in its position and to regulate average stress level. These loading conditions are then verified and validated by numerical simulations.

Verification of loading conditions

The numerical simulation is carried out in Code_Aster with boundary conditions as defined in Fig. 2.20. Because of geometrical symmetry, the FE model is built with 1/8 of the PACIFIC disc. Pressures are applied to different surfaces to represent the compressive

Limitations of PACIFIC test

According to the above simulations, it is concluded that fatigue crack propagation under LSY condition can be performed with the PACIFIC set-up. However, the utilisation of the PACIFIC set-up is quite restrained in the study the mechanism of LSY fatigue crack growth for a few reasons. The most important reason is that the duration of the test is too long. It takes 90 minutes to fulfill a PACIFIC cycle, a month is at least required to conduct 500 cycles without taking into account of the necessary interruptions during test. Other reason consists in the limitation of observation. The PACIFIC instrumentation does not allow a closer observation of crack tip region, which is mostly required for the understanding of fatigue crack growth mechanism; Therefore, due to these reasons, a simplified mechanical fatigue test is of necessity. This mechanical test should avoid the above limitations and most importantly should be able to produce LSY conditions with comparable mechanical field as observec in PACIFIC disc in order to study the LSY fatigue crack propagation problem.

Conclusion

• An isothermal loading condition to create large scale yielding conditions in PA-CIFIC disc is defined and verified by elastic / elastic-plastic FE simulation.

• A decreasing strain field along crack growth direction is observed in the elasticplastic simulation of PACIFIC test, which implies that the stable LSY fatigue crack propagation can be performed on PACIFIC disc.

• Analysis of the limitations of PACIFIC test make it essential to propose a simplified mechanical set-up which allows LSY fatigue crack propagation so as to better understand the LSY fatigue crack propagation mechanism.

Development of a simplified test for fatigue crack propagation in LSY condition

This section deals with the development of a fatigue test protocol, which should make it possible to perform stable fatigue crack growth experiment in large scale yiel-ding conditions under mechanical loadings. The proposed mechanical specimen is introduced in the first place, followed by the presentation of experimental set-up on the next. The geometry is validated by FE analysis in the end.

Specimen geometry

The test aims at firstly, enriching the experimental data of fatigue crack propagation in LSY conditions in order to better understand the mechanism of crack propagation in this situation; secondly, the mechanical field created in the mechanical specimen is required

Finite element simulation of the mechanical fatigue test

The FE analysis of mechanical fatigue test is performed with Code_Aster. Objective of this section is firstly to identify the experimental boundary condition; and secondly to validate the specimen geometry in the creation of LSY conditions.

Identification of the boundary conditions

The misalignment in the SENT-type specimen imposes a bending moment applied by the MTS machine on the specimen. This bending moment is determined by the stiffness of the loading axis of the machine. Thus the identification of the boundary condition consists in the definition of the machine's stiffness.

To do so, such a FE model is created as shown in Fig. 2.32. Half of the specimen is modelled because of geometrical symmetry. The MTS machine is represented by point M, which is connected to the specimen. Symmetric and mechanical boundary conditions are applied as shown in this figure. The cyclic force-controlled loading of pre-crack step is applied at point M along axis y with a maximum force of 11kN and a loading ratio of 0.1. This loading ensures that the specimen deforms elastically. Elastic simulation has been performed with machine's stiffness ranging from 1N • m to 10.E5N • m. The displacements of two points E25 and E43 corresponding to the locations of extensometers in the test are recorded and compared with the experimental data. The stiffness of the machine is then determined once the best coherence is found between the numerical and the experimental force-displacement curves.

The best agreement is obtained with the machine's stiffness of 10.E2N •m as displayed in Fig. 2.33, where the continue lines represent the experimental data and the dotted lines for the numerical results. A good coherence is observed between the numerical and experimental force-displacement curves of the point E43 while a slight displacement difference of about 1µm is noted for the point E25.

This difference in displacement could be explained by uncertainty of the extensometer's location during the test. A better agreement can be obtained if the displacement of a nearby point (with x = 12.8mm compared to E25 with x = 11.5mm) is considered in the comparison (Fig. 2.34).

However, the 1.3mm's difference in the location of the extensometer could not be eliminated during the test. Considering that the 1µm's difference between the numerical and experimental displacement is small, it is assumed that this extensometer is always in contact with the specimen at its mid-width (x = 11.5mm). Fig. 2. [START_REF] Facheris | Cyclic mechanical behavior of 316l: Uniaxial lcf and strain-controlled ratcheting tests[END_REF] shows the elastic deformation of the specimen under such a boundary condition. A translation along axis y as well as a bending effect displayed by a rotation along axis z are both observed for the specimen under tensile loading.

Elastic-plastic simulation of the mechanical fatigue test

Elastic-plastic simulation of the mechanical fatigue test give an insight in the mechanical field in the specimen under cyclic load. The fatigue test is under cyclic displacement- This chapter is dedicated to the fatigue crack growth experiments. These tests were performed in order to, on the one hand, obtain and enrich the experiment data of fatigue crack propagation in large scale yielding conditions in 316L SS; and on the other hand, characterize the mechanism of fatigue crack growth in large scale yielding conditions for this material which provides the theoretical fondamentals to the fatigue crack modelling. Two types of experiments were carried out, namely SENT-type experiments under cyclic mechanical loadings and PACIFIC tests under cyclic thermal loadings. This chapter hence includes the presentations of the experimental results of the SENT-type test and the PACIFIC test as well as the comparison between the two tests.

Mechanical LSY fatigue experiments on SENT-type specimen

Mechanical LSY fatigue experiments are carried out on a notched SENT-type specimen (2.25). Pre-cracking experiment under force-controlled cyclic loadings and fatigue crack propagation experiment under strain-controlled cyclic loadings were carried out successively on one specimen. The fatigue crack growth experiments were performed under constant-amplitude as well as variable-amplitude loadings. Load history effect on fatigue crack growth in LSY conditions is exploited by the tests under variable-amplitude loadings. Mechanism of LSY fatigue crack growth is characterized by an analysis of fracture surface under scanning electron microscope (SEM).

Pre-cracking

In order to get rid of the notch's influence on the crack growth, a pre-crack was created for each specimen before the fatigue test. A pre-crack of 1mm is chosen which is twice the notch's width. The pre-crack was performed under a force-controlled cyclic loading with a maximum load of 11kN and a loading ratio of R F = 0.1. Loading frequency was set at 20Hz. The specimen remains in its elastic domain during the pre-cracking test.

During the test, a surface replication method was adopted in the detection of crack initiation and the pre-crack length measurement. An extensometer was employed to record the crack tip opening displacement (CT OD). It was found that a critical variation of 2µm in CT OD corresponds to a crack length of about 1mm. This observation greatly facilitates the pre-cracking procedure. Instead of measuring the pre-crack length by the replication method which requires an interruption of the on-going test and usually takes a lot of time, the pre-crack length can be known continuously by the variation in the ∆CT OD.

LSY fatigue crack growth 1.2.1 Test loading conditions

Once the pre-cracked specimens were prepared, a series of experiments under different loading conditions was performed. Constant-amplitude as well as variable-amplitude loading conditions were applied as shown in Table 3.1. Three constant-amplitude loadings are defined with maximum strains of 0.3%, 0.5% and 0.7% respectively. Two variableamplitude loadings are defined with 100 cycles of high load with a maximum strain of 0.7% followed by a low constant-amplitude load with a maximum strain of 0.3% or 0.5% (Fig. 3.1). The strain-controlled cyclic loading tests were monitored by an extensometer with a measurement range of 25mm (Fig. 2.28). A strain ratio of 0.2 was fixed for all amplitudes. Loading frequency was set at 0.1Hz. Pictures were taken every 50 cycles at maximum load for the crack length measurement. A few conclusions can be drawn from this graph: firstly, stable fatigue crack propagation was observed for all the tests. The crack growth rate, representing by the slope of each fatigue crack growth curve, tends to decrease with the number of cycles, implying a diminution of crack growth rate. Secondly, the crack growth rate under the same loading conditions are in good coherence demonstrated by two parallel curves: the gap between the curves is due to the difference in pre-crack lengths, indicating that the tests are reproducible. Thirdly, it could be observed that the crack growth rate increases with maximum strain. Fourthly, it is found that the force ratio R F defined by F max /F min decreases during the test and the force ratios for the three loading amplitudes are almost identical as shown in Fig. 3.3. The global crack growth rate da/dN is obtained by a linear regression method over 5 successive experimental points. The crack growth rate for different loading conditions is plotted in Fig. 3.4. From this graph, the decreasing crack growth rate could be easily observed for each test. The crack growth rate increases with maximum load, ranging from 3µm per cycle for ε max = 0.3% to as much as 12µm per cycle for ε max = 0.7%.

The global crack growth rate da/dN from 3µm per cycle to 12 µm per cycle is quite unusual compared to the crack growth rate in small scale yielding (SSY) conditions for 316L SS. Fremy [START_REF] Fremy | Fissuration par fatigue en mode mixte I+II+III non proportionnel dans l'acier 316[END_REF] has observed a crack growth rate of less than 1µm per cycle in SSY condition for the same material. This phenomena makes it essential to explore the mechanism of fatigue crack propagation in LSY loading conditions.

Fractographic analysis Fractographic analysis is a useful tool in the study of fracture mechanism and it was also adopted in the exploration of fatigue crack growth mechanism in LSY conditions. Fig. 3.5 shows a global fracture surface of specimen after constantamplitude loading of ε max = 0.7%. It is observed that under constant-amplitude cyclic loading, the crack path remains coplanar. Three zones can be distinguished from the • Plastic zone observation

• Fracture mechanism analysis

Preparation of PACIFIC specimens

According to different purpose, various types of specimens are prepared. Fig. 3.23 shows the cut-plan of PACIFIC disc in the preparation of small specimens. CT type specimens for C1 and C4 are prepared for the study of fracture mechanism by examining the fracture surfaces. C3 is used for the crack path observation on both top and bottom surfaces (labelled as C3-top/bottom). The two other cuts, one is at small radius (labelled as SRtop/bottom) and the other at large radius (labelled as LR-top/bottom) of the mock-up is for the plastic zone observation. The lateral faces of CT type specimens are polished for crack length measurements before they are opened under mechanical cyclic loadings. A mechanical polishing followed by a polishing of colloidal silica equivalent to 0.5µm are performed on other specimens for SEM observation.

Analysis of crack growth path and crack length

This analysis is performed on specimen C3. Crack observation with SEM is performed on both top and bottom surfaces of the specimen. Fig. 3.24 show the crack path of C3 on top surface. The crack is composed of a pre-crack generated under cyclic mechanical loading and a fatigue crack produced by cyclic thermal loading. It could be observed that the thermal fatigue crack is less straight-lined and more secondary cracks appear along the crack path compared to the pre-crack. Intra-granular cracking is observed for precrack while it is difficult to tell the cracking mode for thermal fatigue crack with this observation. As a matter of fact, the grains around crack path are deformed too heavily to be distinguished as shown in Fig. 3.25. A further EBSD analysis is done to illustrate the plasticity around crack path. Surface crack length obtained under SEM is compared to the measurement given by an optical microscope (Table .3.9). About 100 µm's difference is observed between the • Important plastic zone was observed around the crack path, indicating LSY fatigue crack growth of PACIFIC test.

• Striation were observed on fracture surface of PACIFIC specimen. Good consistency is obtained between the striation spacing and da/dN. The striation crack growth mechanism is proved to be the dominant reason for fatigue crack propagation in LSY conditions.

Comparison between PACIFIC and SENT tests

Firstly, both stable fatigue crack growth tests in large scale yielding conditions were performed on SENT-type specimen and PACIFIC disc and good agreement in the crack growth rates is obtained for the two tests. For the test under mechanical loadings, the crack growth rate decreases with crack length for a given load amplitude. A crack growth rate ranging from 12 µm per cycle to 0.5 µm per cycle is obtained as shown in Fig. 3.35.

For PACIFIC test, a slight increase in crack growth rate is observed with crack length.

A crack propagation rate ranging from 5µm to 12 µm per cycle is obtained as shown in Fig. 3.36.

Secondly, striation mechanism is found to be responsible for the LSY fatigue crack growth in both mechanical and thermal tests. Striations are observed on the fracture surfaces of both tests and the striation spacing is found to be in good consistency with microscopic crack growth rate da/dN for both tests as well.

To summarize, the test under mechanical loadings is proven to be equivalent to PA-CIFIC test. Thus in the following, the crack growth model is identified with experimental results of mechanical test before it is used to interpret the PACIFIC test.

Conclusions

The fatigue crack growth in large scale yielding conditions are highlighted in this chapter. Two types of tests have been performed: one fatigue test with SENT-type specimen under mechanical loadings and the other on PACIFIC set-up under thermal loadings. Several conclusions could be drawn here:

Stable fatigue crack growth in large scale yielding conditions are achieved under both mechanical and thermal loadings. Crack growth rate increases with loading amplitudes and a decreasing crack growth rate was observed under a given loading amplitude. Good consistency is found in the crack growth rates of mechanical test and PACIFIC test, with the crack growth rating ranging from 0.5 µm per cycle to 12 µm per cycle on SENT-type specimen and from 3 µm per cycle to 10 µm on PACIFIC disc. This growth rate is quite unusual compared to the crack growth rate in SSY condition for this material which is usually less than 1 µm per cycle.

Striations are observed on the fracture surfaces of both SENT specimen and PACIFIC specimen. Large striation spacing is obtained which is in good coherence with the macro-scale yielding conditions. This observation makes it necessary to take account of this effect in the LSY fatigue crack growth prediction, even though the plasticity is no longer confined in the crack tip region in large scale yielding condition.

The conclusion of striation caused crack growth in LSY condition is essential for that it provides the evidence of the application a fatigue crack growth model in this study. This fatigue crack growth model is based on the hypothesis that crack propagates due to the creation of striation. The development and application of this model in the prediction of SENT and PACIFIC crack growth will be presented in the following chapter. This chapter is devoted to the extension of the crack propagation model initiated at LMT laboratory of ENS Cachan, from SSY conditions to LSY conditions. To do so, this chapter is divided into four parts: a brief introduction of the incremental model will be given in the first place, mainly including the hypothesis on which the model is based; Then, the development of this model which is composed of a propagation law and a plastic blunting law from SSY to LSY conditions will be highlighted separately in the second and third parts. In each development, the material hardening effect is also analyzed. The last part draws the conclusion for this chapter.

Introduction 1.Mechanism of crack propagation

According to the models proposed by Neumann [START_REF] Neumann | Coarse slip model of fatigue[END_REF], Laird [START_REF] Laird | The influence of metallurgical structure on the mechanisme of fatigue crack propagation[END_REF], Pelloux [START_REF] Pelloux | Mechanisms of formation of ductile fatigue striation[END_REF] and Li [START_REF] Li | Vector ctd criterion applied to mixed mode fatigue crack-growth[END_REF], fatigue crack propagation is assumed to stem from a blunting and resharpening mechanism resulting from plastic deformation at crack tip. Reinhard Pippan [START_REF] Pippan | On the mechanism of fatigue crack propagation in ductile metallic materials[END_REF] has found that this crack growth process is independent of small-or large-scale yielding, which is also confirmed by our experimental observation in large-scale yielding condition. A linear relation between the macroscopic crack propagation rate (da/dN) and the CTOD [START_REF] Pippan | Fatigue crack closure: From lcf to small scale yielding[END_REF] has also been observed by many researchers in either experimental or numerical approaches in ductile materials. In order to apply this relation to variable-amplitude loadings, a time derivation rather than a cyclic derivation is preferred (Eq.4.16) [START_REF] Pommier | Time-derivative equations for mode i fatigue crack growth in metals[END_REF][START_REF] Pommier | Incremental model for fatigue crack growth based on a displacement partitioning hypothesis of mode i elastic-plastic displacement fields[END_REF]:

da dt = α dρ dt (4.1)
where ρ is viewed as the displacement between the crack flanks induced by plastic deformation (crack tip opening or sliding displacement) as depicted in Fig. 4.1. The newly created crack area be expressed as follows:

ȧn * = α(t ∧ ρ) (4.
2)

It provides the rate ȧn * of production of cracked area ȧ per unit length of the crack front t along a growth plane defined by its normal n * , as a function of the crack tip displacement rate. Due to the fact that the crystal orientation and the slip plane are not taken into consideration by this model, the parameter α is empirical and has to be identified with mode I fatigue crack propagation test results.

Hypotheses

The crack is assumed to be locally planar and in generalized plane strain conditions along the local straight crack front. With this assumption, it is possible to attach a local coordinate system R T (w,n,t) to a point T of the crack front as illustrated in Fig. 4.2. Then the

Scale-invariance

It is assumed that the radius of crack blunting is sufficiently small and negligible compared with the characteristic scales of the problem (crack length). With respect to this hypothesis, the geometry of the crack is regarded as locally scale invariant. This hypothesis implies that the local velocity field v(P) could be represented in a local scale invariant form. Such a form could be chosen as a product of a function of scale f (r) (where r represents the distance to the crack tip) and a function g(θ) (where θ represents the angular position with respect to the crack plane) as shown in Eq.4.4.

v(r,θ) R T = f (r)g(θ) (4.4)
It is worth noting that the analytical solutions of the mechanical field within the crack tip region provided by Westergaard [START_REF] Westergaard | Bearing pressures and cracks[END_REF] in linear elasticity display the same characteristics. It is also possible to obtain the numerical field within the crack tip region from FE simulation and express the field in the form in Eq.4.4 with the help of the algorithm of Karhunen-Loeve [START_REF] Karhunen | Über lineare methoden in der wahrscheinli-chkeitsrechnung[END_REF].

Algorithm of Karhunen-Loeve(KL)

The algorithm of Karhunen-Loeve and other similar proper orthogonal decomposition (POD) methods are widely used to compress images or films. They are also applied in fluid or solid mechanics as a reduction model technics. In our study, the Karhunen-Loeve decomposition is used to extract essential components of a movement so as to model it in a simplified manner. To achieve this, this method is applied to decompose a space-time function into a sum of products of a spatial function (reference field) and a time function (intensity factor); each intensity factor corresponds to a degree of freedom.

Assume that the displacements (U) of a set of nodes (N) are known for a set of instants (T ). The spatial-time matrix V can be constructed as in Eq.4.5. The spatial self-correlation matrix C constructed as in Eq.4.6 is defined for the purpose of extracting essential components of the movement:

V (u N ,T )=    U (n=1,t=1) ... U (n=1,t=T ) . . . . . . . . . U (n=N,t=1) ... U (n=N,t=T )    (4.5) C = V •V T (4.6)
The eigenvectors of the matrix C, denoted F k , form the basis of spatial movement fields. Since eigenvectors are orthogonal one to another, they correspond to the basic mobilities of the system.

The time coefficients (intensity factors) A k are obtained by projecting the matrix V onto each eigenvector F k (Eq.4.7):

A k = V T • F k (4.7)
Thus, the Karhunen-Loeve decomposition of the matrix V can be written as a sum of products of eigenvectors and the corresponding intensity factors (Eq.4.8):

V = n ∑ k=1 A k • F k (4.8)
An approximation of the matrix V can be made by considering only the first b modes of the decomposition,

e V = b ∑ k=1 A k • F k (4.9)
The global error associated with this approximation as a function of the number of decomposition modes used to represent the movement is calculated as in Eq.4.10. This error drops down when the number of modes used in the approximation increases, which at the same time adds additional degrees of freedom. The calculation of the relative error between the finite element field and the approximate field is required to limit the number of degrees of freedom.

relative error = s ∑ i,k (V ik -e V ik ) 2 ∑ i,k (V ik ) 2 (4.10)
If additional modes in the decomposition do not reduce the error. It can be assumed that the approximation contains enough degrees of freedom.

Application of Karhunen-Loeve in an elastic case In order to illustrate this method, the algorithm Karhunen-Loeve is applied to the post-treatment of an elastic finite element calculation. Here, the V (u N ,T ) corresponds to the displacement field u e I (r,θ) obtained for K I = 1MP √ m of each node in the crack tip region, with (u N , T ) referring to coordinate position (r,θ) of each node in the local frame R T .

If only the first order of Karhunen-Loeve decomposition is taken into consideration, this method allows representing the displacement field from FE analysis as a product of a function of f (r) and a function of g(θ) as in Eq.4.4. The error associated with this approximation is rather small: 0.05% for 3078 nodes located between 50µm < r < 1000µm around the crack tip.

The analytical solution of the displacement field within the crack tip region in linear elastic conditions provided by Westergaard is also expressed as a product of a radial function and an angular function. The comparison between the numerical and the analytical solutions is plotted in Fig. 4.3 and Fig. 4.4. The angular distribution of displacement is illustrated by a circle with radius = 1 which is deformed under effect of g(θ) 1 or Westergaard solution. Good agreement is obtained. It is worth noting that with the Karhunen-Loeve decomposition, the initial problem (displacement vector of 3078 nodes) is greatly condition is labelled as u g . As a result, the velocity field in crack tip zone in LSY condition is approximated by Eq.4.14.

v(P,t)= K(t)u e (P) | {z } v e (P,t) + ρ(t)u c (P) | {z } v c (P,t) + ġ(t)u g (P) | {z } v g (P,t) (4.14)
If the material behavior is linear elastic, then the intensity factor K of the elastic part of the velocity field is equal to the nominal stress intensity factor K∞ . Otherwise, these two quantities are slightly different, because elastic strain may arise not only from applied stresses (and therefore from K∞ ) but also from internal stresses caused by the crack tip plasticity. The difference ( K -K∞ ) can be interpreted as the shielding effect of the plastic zone [START_REF] Rice | Ductile versus brittle behavior of crystals[END_REF].

This approximation (Eq.4.14) is performed by using Karhunen-Loeve algorithm [START_REF] Rao | The Transform and Data Compression[END_REF]. Therefore the reference fields (u e , u c , u g ) are considered as kinematically independent, leading to Eq.4.15.

Incremental model

With respect to the above hypotheses, the incremental model is composed of two laws:

• A propagation law based on the mechanism of fatigue crack propagation which gives the incremental crack growth rate being proportional to plastic blunting rate (Eq.4.16). In this equation, α is a parameter to be identified with the experimental data.

da dt = α dρ dt (4.

16)

• A plastic blunting law of the model which describes the elastic-plastic behavior in crack tip region as presented in Eq.4.17. It provides the plastic blunting rate as a function of the applied nominal stress intensity factor rate and the internal variables (V int ).

dρ dt = f (dK I , K I ,V int ) (4.17)
The objective of this chapter is to enrich this model in order to predict the crack growth rate in LSY condition. Experimental results have shown that equivalent fatigue crack growth mechanism is applied in LSY conditions as well as in SSY condition, implying the same form of propagation law in LSY condition. Numerical analysis will be performed in this chapter to validate this conclusion. The plastic blunting law is studied by FE analysis as well and a series of equations in relation to internal variables should be proposed in consideration of LSY condition.

Extension of the crack propagation law to LSY conditions

This part is devoted to the effect of LSY on the propagation law of the incremental model by a numerical approach. Due to the fact that the crack propagation is caused by the plasticity of the crack tip region and the striation mechanism has been proved, by the experiment results, being the dominant cause of fatigue crack growth despite the occurrence of general plastic flow in previous chapter, the elastic-plastic velocity field in the crack tip region is taken into examination in this study. It also leads to the application of a simplified crack front model in the simulations. Structural effect is not taken into account in the present study.

Numerical model

Such a simplified 2D finite element model is used in the analysis of the elastic-plastic behavior of crack tip region as illustrated in Fig. 4.7. The model is a quasi-infinite sheet with a dimension of 2m×2m and a through thickness central crack in length of 2a = 20mm.1/4 of the model is simulated because of the geometrical symmetry. Linear plane strain elements are employed. A refined and structural mesh is imposed to a circular zone (named D) with 50µm < radius < 1mm around the crack tip. Symmetric boundary condition are defined to the edges concerned. Uniaxial displacement loading is applied. The displacements of the nodes in zone of interest (D) are recorded at every time increment throughout the computation for the following post-treatment.

In SSY conditions, the stress intensity factor K I can be easily determined by the linear relation of u y = K I • u * y with u y corresponding to the nominal displacement loading and u * y referring to the loading which allows introducing a stress intensity factor of K I = 1MPa • m 1/2 . This relation permits the control of the numerical loading by K I rather than by u y . However, it is no longer available once large scale yielding takes place. Considering the plane strain condition and the Von Mises plastic criteria, LSY occurs when a critical value of

K crit I = 2σ ys 1-v-v 2 • √ πa is obtained, beyond which, the K I is calculated from the relation of K I = σ yy √
πa where σ yy refers to the nominal crack opening stress far from the crack tip region.

Development with an ideally elastic-plastic material

In order to analyze firstly the effect of LSY conditions in the crack propagation law with no additional effect related to the material hardening, a material with an ideally elasticplastic behavior is used in FE simulations. The LSY effect is explored by analyzing the reference fields in the approximation of the velocity field of crack tip zone.

d K(t)=

∑ P∈D (du EF (P,t))u e (P) ∑ P∈D u e (P)u e (P) (4. [START_REF] Clerivet | Study of crack tip opening under cyclic loading taking into account the environment and r ratio[END_REF])

du e (P,t)=d KI (t)u e (P) (4.20)
The residual incremental displacement is defined in the following manner:

du residu (P,t)=du EF (P,t) -du e (P,t) (

The residual displacement is a correction to take in the approximation of incremental displacement field in order to take account of the plastic deformation in the crack tip region. Considering the scale invariance hypothesis of crack geometry, each independent component of the velocity field can be expressed by a product of an intensity factor and a spatial distribution. This decomposition of du residu is performed with the Karhunen-Loeve algorithm. By extracting the first term of Karhunen-Loeve decomposition, the following equation is obtained:

du residu (P,t)= f (t)A(P) (4.22)
The spatial distribution A(P) is then dimensioned by a multiplication factor k 0 so that the intensity factor ρ of the reference field u c can be interpreted as the crack tip opening displacement (CT OD). To do so, the Karhunen-Loeve algorithm is applied again to the spatial distribution A(P)=A(r,θ) in order to express it as a product of a radial dependence p(r) and an angular dependence of q(θ). The q(θ) is dimensioned by a coefficient k to ensure that q y (π) -q y (-π)=1, thus we obtain A(P)=g(θ) • f (r) with g(θ)=q(θ)/k and f (r)=kp(r). The radial function f (r) is fitted by an exponential equation f (r)=k 0 exp(-2p a r). Then the small-scale yielding reference field u c is obtained by u c = A(P)/k 0 . The method is precisely illustrated in Fig. 4.8.

The radial and angular dependences f (r) and g(θ) are plotted in Fig. 4.9. It is observed that the radial dependence f (r) decreases with the distance from the crack tip, implying that the plasticity is well confined in crack tip region. The angular dependence is represented by a deformed circle with r = 1. Displacement discontinuity across the crack faces which corresponds to the CTOD can be read directly from the figure .   Large-scale yielding reference fieldu g The same procedure is employed in the construction of the spatial reference field of large-scale plasticity u g . An elastic-plastic numerical simulation with linear displacement loading from 0 to K max is performed. The applied K max is chosen to generate general plastic flow. The displacements of the nodes in domain D are recorded as well during the simulation (u EF (P,t)).

The rate of the intensity factor dρ(t) for small-scale yielding is calculated by Eq.4.23. The incremental elastic and the small-scale yielding displacement fields are approximated by the products of the reference field and corresponding intensity factor according to Eq.4.20 and Eq.4.24. Thus the residual elastic-plastic velocity field obtained by Eq.4.25 du residu (P,t) • A third reference field u g proposed to account for the large-scale plasticity is found necessary and sufficient in the approximation of the velocity field of the crack tip region in presence of large scale yielding conditions.

1 POD spatial / time g(P) m(t) 2 POD r / θ p(r) q(θ) 3 Dimensionning of g(θ) g(θ = π)= q(θ = π) k = 1 2
• Hypotheses of infinitesimal deformation and scale invariance of the incremental model make it possible to express the u g as a product of a radial function and an angular function, namely u g = f (r)g(θ) with the method of Karhunen-Loeve decomposition. Materials with/without hardening behavior have been used in the extraction and spatial distribution analysis of u g . Similar observations have been obtained: f (r) decreases when r reduces and g(θ) shows no discontinuity across the crack faces.

• Fatigue crack growth tests have demonstrated and confirmed the striation crack mechanism in large scale yielding conditions, implying that the fatigue crack propagation is a consequence of plastic blunting and resharping process. The fact of no discontinuity across the crack faces observed for g(θ) leads to the conclusion that the large scale plasticity has no direct contribution to the crack propagation. In other words, the crack propagation law is merely related to the intensity factor of small-scale yielding (ρ) which is assumed to represent the crack tip opening displacement by its definition. Thus the crack propagation law can be expressed as ∝ dρ/dt.

Extension of the plastic blunting law in LSY conditions

As has been discussed previously, the elastic-plastic behavior of the crack tip region can be described by the three intensity factors K, ρ and g once the corresponding reference fields are defined a priori. It is also observed that the fatigue crack propagation is only related to the intensity factor of small-scale yielding reference field ρ, known also as plastic blunting. Thus in order to predict the crack propagation rate, it is essential to determine the plastic blunting rate, dρ/dt. The evolution of dρ/dt is defined as plastic blunting law.

Taken an arbitrary cyclic variable-amplitude loading for example (Fig. 4.14), the behavior of the crack tip region expressed by the Kρ curve which is obtained from the post-treatment of numerical simulations can be divided into two categories: a monotonic behavior with increasing maximum loading and a cyclic one beneath the monotonic loading. This behavior is analogue to that of material cyclic constitutive behavior with monotonic behaviour similar to isotropic hardening and cyclic behavior to kinematic hardening. Therefore, analogue internal variables are proposed in the description of the "constitutive behavior" of a crack. The plastic blunting law is expressed in function of nominal stress intensity factor as well as the internal variables (Eq.4.17).

Each behavior is characterized by its center and size of the elastic domain as shown in

Influence of LSY condition on monotonic behavior of plastic blunting law

Monotonic behavior of plastic blunting law is characterized by the size (2K m ) and the center (K xm ) of the monotonic elastic zone as shown in Fig. 1.1. Previous study has demonstrated the evolution of the two internal variables in SSY condition. In our study, it is expected to enrich their evolutions in LSY condition. Similar to the approach adopted in the cyclic behavior study, influence of large-scale plasticity on monotonic behaviour is analysed by comparing the monotonic behavior is SSY condition. An ideally elastic plastic material is used in the first place before material hardening effect on monotonic behaviour is examined later.

Monotonic behaviour with ideally elastic plastic material

Comparative simulations as illustrated in Fig. 4.23 are performed so as to study the influence of large scale yielding condition on monotonic behaviour of plastic blunting law. One simulation is in SSY condition and the other is under LSY condition. In the first place, an ideally elastic plastic material is used. Simulations with different maximum loadings (K max ) from small scale yielding to large scale yielding are performed. The center of the monotonic elastic domain K xm represents the contact point and it corresponds to the stress intensity factor when crack closure happens. Half size of the monotonic elastic domain K m is obtained through Eq.4.32.

K m = |K xm | + K max (4.32)
The post-treatment is carried out for each simulation so as to obtain the corresponding ρ max for each K max . The evolution of K xm and K m in function of ρ max is depicted in Fig. 4.24. The last three points in these figures come from the simulations in LSY conditions, while the other points come from the simulations in SSY conditions. A linear relation is observed between K xm and ρ max in SSY conditions as well as in LSY conditions. Moreover, the two lines coincide with each other. This observation indicates that the relation between K xm and ρ max in LSY conditions is indifferent with that in SSY conditions. As for the relation between K m and ρ max , the K max saturates when LSY occurs due to the ideally elastic plastic material

Material effect

To verify the above conclusion, different materials are taken into consideration. Materials with linear isotropic or kinematic hardening have been employed.

Here the equations for the internal variables based on the numerical results or theoretical analysis are presented.

Evolution of internal variables for cyclic behavior

Cyclic behavior is characterized by two internal variables (K c and K xc ) representing the size and center of its elastic domain. Therefore it is of necessity to determine the relations of the variables with the plastic blunting ρ as well as the crack length a in order to describe the cyclic behavior of plastic blunting law.

∂K c ∂ρ and ∂K xc ∂ρ

The simulation above illustrates that the size of cyclic elastic domain (2K c ) remains constant if the material presents no isotropic hardening. For materials displaying isotropic hardening, the 2K c is demonstrated to increase with cumulated monotonic plasticity (ρ mono ) as presented in Eq.5.5.

2K c = b c = a √ ρ mono + b (4.33)
where a and b are parameters to be identified by numerical simulations. K xc represents the center of the cyclic elastic domain. So the crack tip region remains elastic (dρ = 0) for an increment dK, if

|K I -K xc | < K c (4.34)
Plasticity takes place in the crack tip region, if:

|K I -K xc | = K c (4.35)
Furthermore, with presence of plastic deformation, the following equation is obtained according to numerical simulation, which is in agreement with Irwin's theory where ρ∝

K I 2 , |K I (t) -K I (t 0 )| = b c + a c p |ρ(t) -ρ(t 0 )| (4.36)
∂K c ∂a and ∂K xc ∂a The size of cyclic elastic domain 2K c is assumed to decrease exponentially with the crack propagation according to Eq.4.37:

∂K c ∂a = -p a K c (4.37)
On the transition from cyclic loading to monotonic loading, we have According to the previous study, the plastic field at the crack tip could be represented in a simplified manner as follows: u c (P,t)=ρ(t)g(θ)e (-2p a r) (4.40) This equation indicates that the plastic deformation decreases exponentially with the distance from crack tip and it is in linear relation with ρ:

K = K xc + K c = K xm + K m ,
ε p θθ (r, θ = 0) ∝ρe (-2p a r) (4.41)
If the size of the domain of confined plasticity is defined by r p at which distance the plastic deformation reaches a critical value ε crit p θθ . In this condition, we have:

ε crit p θθ ∝ρe (-2p a r p ) ⇒ ρ∝ε crit p θθ e (2p a r p ) ⇒ K m 2 ∝ε crit p θθ e (2p a r p ) ⇒ K m ∝ q ε crit p θθ e (p a r p ) (4.42)

Consequently, with derivation by a:

∂K m ∂a = ∂K m ∂r ∂r ∂a = - ∂K m ∂r ∝ -p a q ε crit p θθ e (p a r p ) (4.43)
Taken the first order approximation, the following relation is obtained:

∂K m ∂a = -p a K m (4.44)
K xm represents the center of the monotonic elastic region. In other words, it corresponds to the required loading to be applied in order to compensate the residual stress over the crack faces. Thus, K xm indicates the threshold of crack closure.

Suppose that the residual stress can be conserved and no more plasticity is produced at the crack tip during the crack propagation. The evolution of K xm in relation to the crack length can be obtained by a superposition of two phenomena: firstly, it is calculated that the stress intensity factor of a point load on crack face decreases with the distance to the crack tip and tends to zero. This observation is introduced by Eq.4.45. Secondly, as crack

∂ 2 K m ∂a∂ρ = -p a ∂K m ∂ρ (4.51) ∂ 2 K xm ∂a∂ρ = k b ∂K m ∂ρ -k a ∂K xm ∂ρ (4.52)
Thus, by addition the two equations, we have:

∂ 2 K m ∂a∂ρ + ∂ 2 K xm ∂a∂ρ =(k b -p a ) ∂K m ∂ρ -k a ∂K xm ∂ρ (4.53)
According to Schwarz theory, we have:

∂ 2 K m ∂a∂ρ + ∂ 2 K xm ∂a∂ρ = ∂ 2 K m ∂ρ∂a + ∂ 2 K
8 > > > > < > > > > : ∂K m ∂ρ = k a (k a + k b -p a ) γK IC -γK Im ∂K xm ∂ρ = (k b -p a ) (k a + k b -p a )
γK IC -γK xm = a xm -γK xm (4.55)

Conclusion

In summary, the plastic blunting law giving evolution of internal variables in relation to external loading and crack growth could be concluded as follows: For monotonic behavior:

8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : ∂K m ∂ρ = k a (k a + k b -p a ) γK IC -γK m ∂K xm ∂ρ = (k b -p a ) (k a + k b -p a ) γK IC -γK Ixm = a xm -γK xm ∂K m ∂a = -p a K m ∂K xm ∂a = k b K m -k a K xm (4.56)
For cyclic behavior:

8 > > > > > > > > > > > < > > > > > > > > > > > : 2K c = b c = a √ ρ mono + b |K I (t) -K I (t 0 )| = 2K c + a c p |ρ(t) -ρ(t 0 )| ∂K c ∂a = -p a K c ∂K xc ∂a = ∂K xm ∂a + ∂K m ∂a - ∂K c ∂a (4.57)

Conclusion

This chapter deals with the development of the incremental model in large scale yielding conditions. The development is performed by numerical analysis on crack propagation law and plastic blunting law respectively. A third reference field in account of large-scale plasticity has been proposed and proved useful in the approximation of the velocity field of crack tip region in large scale yielding conditions. However, a spatial distribution analysis of this field found that the large-scale plasticity has no direct contribution to the crack propagation in consideration of striation crack mechanism which is observed in LSY fatigue crack test. Therefore, the crack propagation law is a function of plastic blunting, which is represented by the intensity factor of the small-scale yielding reference field ρ, in other word, the crack propagation rate is proportional to the plastic blunting rate even in presence of large-scale plasticity.

The influence of LSY condition on cyclic and monotonic behavior of plastic blunting law is studied by comparing the difference of evolution of internal variables in both SSY and LSY conditions. Effect of material behavior is studied as well. It is found that the LSY condition has no significant modification to plastic blunting law for any materials. Therefore, the plastic blunting law developed in SSY conditions can be used directly in LSY conditions.

The equations of the plastic blunting model based on numerical observation or theoretical analysis are introduced in the end. These equations describe the evolution of internal variable with crack propagation and crack tip plasticity during external cyclic loadings. The identification of the condensed model will be realized with 316L SS in the following chapter as well as its application in the prediction of fatigue crack growth rate. This part is devoted to the identification and the implementation of the incremental model developed in the previous chapter for the studied material and by using the fatigue test results. An overview of the coefficients in this model is presented in the first place. The identification of the plastic blunting law by numerical simulation and the identification of the crack propagation law based on the experimental data on SENT-type specimen are then presented repectively. The interpretation of the PACIFIC tests with the identified incremental model is performed at last.

Overview of the parameters in the incremental model

The incremental model extended to LSY conditions is proposed in the previous chapter. This model was developped to predict the fatigue crack growth of elastic-plastic materials for complex loading conditions, provided that the coefficients in the model are identified with the suitable material's properties. The coefficients to be identified in the model are listed in the following table, some of them need to be identified by numerical simulations while others come from the experimental results.

Parameters of material properties ∆K th

Fatigue threshold stress intensity factor amplitude K Ic

Fracture toughness The parameters in the plastic blunting law are identified by the FE analysis. The numerical simulation is performed with a 2D model in plane strain conditions with a central through

Incremental updating of the variables

The internal variables K m , K xm , K c and K xc evolve with crack propagation (a) and plasticity (ρ). Evolution in ρ mono , ρ cycl , a mono and a cycl could be observed during one loading increment according to different loading phases (cyclic plasticity or monotonic plasticity). , δK a mono c and δK a cycl c . These variations are calculated according to the equations described in cf. §A. The updating of the variables at the end of each loading increment is performed as follows:

Updating of ρ ρ i+1 = ρ i + δρ cycl + δρ mono (5.12)

Updating de a a i+1 = a i + δa cycl + δa mono Updating of K xc The center of the elastic domain K xc is updated at the end of each load increment and its evolution depends on K I(i+1) .

Loading/opening phase Under opening phase, the center of the cyclic elastic domain follows the loading path. Therefore, we have:

K xc(i+1) = ⇢ K xci si K I(i+1) < K the K I(i+1) -K c si K the ≤ K I(i+1)
(5.17)

As a consequence, under opening phase, K xc remains constant if no plasticity takes place.

Chapter 6

Conclusions and perspectives

The objectives of this thesis were: firstly, to enrich the experimental data of mode I fatigue crack growth in large-scale yielding conditions and analyze the mechanism of the large-scale yielding crack propagation for 316L stainless steel; secondly, to model the large-scale yielding fatigue crack growth by extending the incremental crack propagation model from small-scale yielding conditions to large-scale yielding conditions so as to predict the crack growth in the PACIFIC disc.

To achieve these objectives, the following a methodology was adopted: this work was primarily concentrated on the experimental analysis of fatigue crack growth for 316L SS in large-scale yielding conditions. Two types of experiments under cyclic mechanical and thermal (PACIFIC test) loading conditions have been performed respectively. The mechanisms of crack propagation were explored using fractographic observations. Based on the experimental observation, an incremental crack growth model was proposed. Finite element analysis was performed to study the impact of large-scale plasticity on the incremental model and to develop this model in LSY conditions. The extended incremental model was identified by the experimental results of the mechanical tests and the prediction of crack propagation on PACIFIC disc was performed in the end.

Fatigue crack growth tests in large-scale yielding conditions have been carried out both under constant-amplitude and variable-amplitude mechanical cyclic loading conditions and cyclic thermal loading conditions. Stable large-scale yielding crack propagation has been observed for all the tests. Large fatigue crack growth rates have been obtained for both of the tests with a growth rate ranging from 2µm per cycle to 12µm per cycle, which is much more greater than the crack growth rate in small-scale yielding conditions for 316L SS (less than 1µm per cycle). Load history effect in large-scale yielding conditions has also been observed by a retardation in crack growth rate after an overload.

The mechanisms of large-scale yielding fatigue crack growth has been explored by fractographic analysis. Similar to the fracture surface of fatigue crack growth in smallscale yielding conditions, striations are present as well on the fracture surface of fatigue crack growth in large-scale yielding conditions. Moreover, the striation spacing is found to be in good coherence with the global crack growth rate, demonstrating that the striation mechanism is the dominant reason for the crack growth in large-scale yielding conditions.

A numerical approach was developped to analyze the elastic-plastic behaviour of a representative section of a crack front in large-scale yielding condition. By using a model reduction technique, the velocity field of the crack tip region in large-scale yielding conditions can be approximated by a sum of three components, each component being expressed as a product of an intensity factor used as a degree of freedom and a fixed spatial distribution named reference field. The third field was added, dedicated to capture the large-scale yielding contribution, to the two existing terms for elastic and small-scale yielding contribution. Analysis of the spatial distribution of the large-scale yielding reference field shows no direct influence of large-scale plasticity to fatigue crack growth.

The only "driving force" of the crack growth is the plastic blunting, which is associated to the intensity factor of the small-scale yielding reference field. In other words, the crack growth rate is proportional to the plastic blunting rate presented as da/dt = αdρ/dt. This relation constitutes the crack propagation law of the incremental model. The evolution of the plastic blunting has been explored in large-scale yielding conditions by taking account of different material behaviors. The plastic blunting is in relation with the applied loads and a series of internal variables (dρ = f (d K,V int )). A plastic blunting law for elasticplastic material with isotropic and kinematic hardening in large-scale yielding conditions has been developed.

In small-scale yielding conditions, the representative of loading conditions KI is found to be similar to the K ∞ I . With the extention of the plastic zone, an increase has been observed in the difference between them K and K ∞ I . Therefore, the KI instead of the K ∞ I should be used as the driving force of crack propagation in large-scale yielding conditions.

A stress intensity factor similar to the KI was obtained from the crack opening displacement (COD) for the SENT-type specimen. Compared to the stress intensity factor calculated by the elastic FE analysis (K ∞ I ), the COD-based stress intensity factor is able to take account of both the stresses applied at infinity and the internal stress field in the crack tip region. Crack growth rates estimated by the incremental model were found in good coherence with experimental crack growth on SENT-type specimens.

The identified incremental model was used to predict the crack growth in PACIFIC disc. Underestimated crack growth rate was obtained with the stress intensity factor calculated from the elastic FE simulation. Inappropriate representative of applied loads and complex loading conditions on PACIFIC disc rather than uniaxial loading can be both responsible for this underestimation. A first attempt has been made to approximate a KI so as to re-evaluate the crack growth rate, an overestimation of da/dN was observed, demonstrating that the discrepancy between the experimental results and the predictions of the incremental model could be entirely attributed to the determination of the driving force (K ∞ I or KI )

Several questions were arised during the study, the first question concerns with the representation of driving force in large-scale yielding conditions for the incremental model. In small-scale yielding conditions, difference between the stress intensity factor of the elastic reference field KI and the stress intensity factor K ∞ I remains limited due to the fact that the plasticity is confined in the crack tip region, the K ∞ I can be used in the incremental model to predict the crack growth. However, in large-scale yielding conditions, the difference between the KI and K ∞ I becomes important with the increase of the plastic zone, the K ∞ I can no longer be used in the incremental model in such situations.

Parameters to represent loading conditions should be proposed in large-scale yield-ing conditions. KI is proposed to replace K ∞ I . This parameter could be obtained by the experimental crack opening displacement. However, in situations where the COD is not available, a more general method should be proposed. The difference between KI and K ∞ I is found to be dependent on the plastic blunting ρ and the instensity factor of large-scale yielding reference field g, the ρ dependence can be taken into consideration by the plastic blunting law, however, there is no relation available to take account of the g dependence. Further study should be conducted to explore the g influence on K ∞ I .

Another question concerns about the multi-axial loading conditions. This work is concentrated on the large-scale yielding conditions generated under unixial loadings, the multi-axial loading conditions are not studied in this work. The T stress is observed to have an influence on short crack propagation, thus it may also play a role for long crack growth in large scale yielding conditions. Load history effect have been observed on the fatigue crack growth in large-scale yielding condition. The ability of predicting crack growth under variable-loading conditions by the incremental model should also be verified. Kthe here is different from the threshold of elasticity (K the ). Both of the variables represent the size of cyclic elastic domain, however, Kthe does not evolve during the whole loading until the next elastic domain appears.

Calculation of δa cycl = a i+1 -a i

The incremental crack growth in cyclic plastic domain could be given by the propagation law as follows:

δa cycl = α |δρ cycl | (A.4)
Calculation of δK a cycl m Equation of K m against a under monotonic plasticity is given as follow:

∂K m ∂a = -p a K m
If this equation is integrated between a i et a i+1 , we have:

K m (a i+1 )=K m (a i ) e -p a (a i+1 -a i )
Therefore, the variation of K a cycl m is given as:

δK a cycl m = K mi ⇣ e -p a δa cycl -1 ⌘ (A.5)
Calculation of δK a cycl xm Equation of K xm in relation to a under monotonic plasticity is given as follow:

∂K xm ∂a = k b K m -k a K xm
Similarly, if this equation is integrated between a i et a i+1 , we have:

K xm (a i+1 )= k b k a -p a K m (a i )e -p a (a i+1 -a i ) + ✓ K xm (a i ) - k b k a -p a K m (a i ) ◆ e -k a (a i+1 -a i )
Therefore, the variation of K a cycl xm is given as:

δK a cycl xm = k b k a -p a K mi ⇣ e -p a δa cycl -e -k a δa cycl ⌘ + K xmi ⇣ e -k a δa cycl -1 ⌘ (A.6)

Monotonic plasticity

Calculation of δρ mono = ρ i+1 -ρ i Equation of δK and δρ under monotonic plasticity is given as follow:

∂K I ∂ρ = γK Ic -γK I
By integrating this equation between ρ i et ρ i+1 , we have:

K I (ρ i+1 )=K Ic +(K I (ρ i ) -K Ic )e -γ(ρ i+1 -ρ i ) (A.7)
Therefore, δρ mono is given as: By integrating this equation between ρ i et ρ i+1 , we have:

δρ mono = 1 γ ln K Ic -K I(i) K Ic -K I(i+1) ! (A.
K m (ρ i+1 )= k a K Ic k a + k b -p a + ✓ K m (ρ i ) - k a K Ic k a + k b -p a ◆ e -γ(ρ i+1 -ρ i )
Therefore, the variation of K

  Temperature distribution on mock-up .................... 87 3.18 Strain distribution on PACIFIC's mock-up ................. 88 3.19 Boundary conditions for PACIFIC simulation ............... 89 3.20 Validation of numerical simulation ..................... 89 3.21 Illustration of crack growth of C1 on the bottom surface .......... 90 3.22 Evolution of crack length .......................... 90 3.23 PACIFIC sample preparation ........................ 92 3.24 Illustration of a crack path ......................... 92 3.25 Appearance of sliding bands along thermal fatigue crack path ....... 93 3.26 EBSD analysis of plasticity near C3 .................... 93 3.27 Appearance of sliding bands for specimen SR ............... 94 3.28 Illustration of fracture surface ....................... 94 3.29 Illustration of different cracking stage of C4: 1.From pre-crack to thermal fatigue crack 2.Thermal fatigue crack 3. From thermal crack fatigue to final fracture ................................. 95 3.30 Illustration of PACIFIC striation measurement on the fracture surface .. 95 3.31 Striation spacing for C1 ........................... 96 3.32 Striation spacing for C4 ........................... 96 3.33 Comparison between da/dN and striation spacing for C1 ......... 97 3.34 Comparison between da/dN and striation spacing for C4 ......... 97 3.35 Crack growth rate for mechanical tests under constant-amplitude cyclic loadings ................................... 99 3.36 Crack growth rate for PACIFIC test under cyclic thermal loadings .... 99 4.1 Illustration of the crack propagation mechanism ..............103 4.2 R O (x,y,z) : re f erence f rameattachedtothecrackedcomponent.R T (w,n,t) : local f rameattachedtothecracktipT.nisnormaltothelocalcrackplane f rom[23]103 4.3 Comparison between the radial dependence of the analytical solution of Westergaard and the f (r) from post-treatment with POD algorithm of an elastic FE calculation ............................106 4.4 Comparison between the angular dependence of the analytical solution of Westergaard and the g(θ) from post-treatment with POD algorithm of an elastic FE calculation ............................106 4.5 Illustration of elastic-plastic separation ...................107 4.6 Evolution of relative errors [43] ......................108 ...............................

Figure 1 . 9 :

 19 Figure 1.9: Schematic cyclic stress-strain behavior ahead of a fatigue crack

Table 1 . 1 :

 11 Chemical composition of 316L

∑

  P∈D u e (P)• u c (P)=0, ∑ P∈D u e (P) • u g (P)=0, ∑ P∈D u c (P) • u g (P)=0,(4.15)

4 gFigure 4 . 8 :

 448 Figure 4.8: Method for the obtention of u c

Fig. 1 . 1 .

 11 Thus K xc and K c are defined for the center and size of the elastic domain of cyclic increases linearly with √ ρ mono with b c = a √ ρ mono + b where a and b are the fitted parameters.

Parameters to be identified by numerical simulation p a

 a Parameter representing the exponential decay of the SSY plastic field in the crack tip region k a & k b Parameters used in the evolution law of the crack opening level K xm with respect to a crack extent without plastic strain γ Parameter used in the evolution law of K m versus plastic blunting ρ a c Parameter used in the evolution law of K xc versus ρ a, b Parameters used in the evolution law of K c versus ρ mono Parameters to be identified by experimental results α Parameter used in the crack propagation law 1.2 Identification of the parameters of the plastic blunting law for 316L SS

(5. 13 )( 5 . 14 )( 5 . 15 )

 13514515 Updating of K m and K xm K m(i+1) = K mi + δK a cycl m + δK ρ mono m + δK a mono m K xm(i+1) = K xmi + δK a cycl xm + δK ρ mono xm + δK a mono xm Updating of K c K c(i+1) = K ci + δK a cycl c

=

  one increment is estimated by the following equation: K ci (e -p a δa -1)(A.2) where δa could be δa mono or δa cycl .Calculation of δρ cycl = ρ i+1 -ρ iUnder the phase of cyclic plastic conditions, we have:|K I (ρ i ) -K I (ρ 0 )| = a c p |ρ i -ρ 0 | + b c et |K I (ρ i+1 ) -K I (ρ 0 )| = a c p |ρ i+1 -ρ 0 | + b cδρ = ρ i+1 -ρ i could be obtained by combining the two equations:δρ cycl = Sign(K I(i+1) -K I(i) )

8 )

 8 Calculation of δamono = a i+1 -a i δa mono = α |δρ mono | (A.9)Calculation of δK ρ mono mEquation of K m in relation to ρ under monotonic plasticity is given as follow:∂K m ∂ρ = k a k a + k b -p a γK Ic -γK m

1 ⌘ (A. 12 )

 112 k a + k b -p a -K mi ◆ ⇣1 -e -γ δρ mono ⌘ (A.10)Calculation of δK ρ mono xmEquation of K xm against ρ under monotonic plasticity is given as follow:∂K xm ∂ρ = k b -p a k a + k b -p a γK Ic -γK xmBy integrating this equation between ρ i et ρ i+1 , we have:K xm (ρ i+1 )= k b -p a k a + k b -p a K Ic + ✓ K xm (ρ i ) -k b -p a k a + k b -p a K Ic ◆ e -γ(ρ i+1 -ρ i )Therefore, the variation of Kρ mono xmis given as:δK ρ mono xm = ✓ k b -p a k a + k b -p a K Ic -K xmi ◆ ⇣ 1 -e -γ δρ mono ⌘ (A.11)Calculation of δK a mono m Equation of K m in relation to a under monotonic plasticity is given as follows:∂K m ∂a = -p a K mBy integrating this equation between a i et a i+1 , we have δK a mono m given in following expression:δK a mono m = K mi ⇣ e -p a δa mono -Calculation of δK a mono xmEquation of K xm in relation to a under monotonic plasticity is given as follow:∂K xm ∂a = k b K m -k a K xmBy integrating this equation between a i et a i+1 , we have δK a mono xm given by the following expression: a δa mono e -k a δa mono ⌘ + K xmi ⇣ e -k a δa mono -1 ⌘ (A.13)

  .................................. 1.35 Load history effect on 316L SS at room temperature ............ 1.36 Evolution of the stress amplitude during increasing and decreasing strain amplitude tests on 316L at three temperature levels [62] .......... Hysteresis loops for variable-amplitude loading fatigue test at 20 • C ... 2.4 Stress for constant-amplitude loading fatigue test at 20 • C ......... 2.5 Influence of temperature on the maximum stress of constant-amplitude loading fatigue tests ............................. 2.6 Evidence of history effect at 150 • C ..................... 2.7 Evidence of history effect at 300 • C ..................... 2.8 Stabilized cyclic stress-strain curves: comparison between experimental and numerical results at room temperature ................. 2.9 Stabilized cyclic stress-strain curves: comparison between experimental and numerical results at 150 • C and 300 • C ................. 2.10 Simulation of history effect at 150 • C .................... 2.11 PACIFIC set-up ............................... 2.12 Pre-cracking procedure: 1. Initial PACIFIC plate; 2. Mechanical fatigue test for pre-cracking ............................ 2.13 Preparation of PACIFIC disc ........................ 2.14 Example of instrumentation scheme for the mock-up ........... 2.15 Example of instrumentation scheme for the complementary ring ..... 2.16 Final PACIFIC disc ............................. 2.17 Principle of the PACIFIC bench: (1) Water and oil circuits. (2) Application of thermomechanical loading on PACIFIC specimen ......... 3.12 Comparison of crack propagation rate between variable-amplitude loading and constant-amplitude loading at ε max = 0.5% ............ 83 3.13 Comparison of crack propagation rate between variable-amplitude loading and constant-amplitude loading at ε max = 0.3% ............ 83 3.14 Evolution of the LSY fatigue crack length with or without overloads in 316L ..................................... 84 3.15 Illustration of pre-cracks on PACIFIC disc ................. 85 3.16 Radial thermal gradient in complementary ring .............. 86 3.17

2.1 Specimen geometry of FFL2 ........................ 2.2 Loading conditions for the LCF tests: 1. constant-amplitude cyclic loading; 2. variable-amplitude cyclic load with increasing strain amplitudes at 20 • C & variable-amplitude cyclic load with increasing and decreasing strain amplitudes at 150 • C and 300 • C; ................... 2.3

Table 2 . 1 :

 21 Chemical composition of 316L SS in use

	Mass %	C	Mn	Si	S	P	Ni	Cr	Mo	Cu	N
	-	0.026 1.81 0.42 0.004 0.033 12 16.96 2.43 0.03 0.05

Table 2 . 2 :

 22 Thermal characteristics of 316L SS[2]

	Temperature	Thermal expansion coefficient	Thermal conductivity	Volumetric heat capacity	Thermal diffusivity

Table 2 . 6 :

 26 Identified material parameters

		Parameters 20 • C	150 • C 300 • C
	Elasticity	E (MPa) ν	190000 170000 160000 0.3 0.3 0.3
		C 1 (MPa)	90000 117907 170000
	Kinematic Hardening	γ 1 C 2 (MPa)	630 9000	1200 17000	1500 25000
		γ 2	100	150	200
	Isotropic Hardening	R 0 (MPa) b	180 5	125 5	110 10
		µ	12	16	16
	History Effect

  • C to 220 • C and constant thermal load at 40 • C, a cycle takes 90 minutes. It can take as much as 2 hours to accomplish a cycle under non-isothermal condition. Therefore, a large sum of time is required to perform one PACIFIC test.

  PACIFIC thermal LSY experiments on a quasi-structure specimen ... 85 2.1 Preparation of PACIFIC test ..................... 85 2.2 Monitoring of PACIFIC test ..................... 85 2.3 Post-treatment of PACIFIC test ................... 90 2.4 Conclusion of PACIFIC test ..................... 97 Comparison between PACIFIC and SENT tests .............. 98 Conclusions .................................. 98

	Chapter 3
	Fatigue experiments in Large Scale
	Yielding (LSY) conditions

Contents

Mechanical LSY fatigue experiments on SENT-type specimen ..... 74 1.1 Pre-cracking ............................. 74 1.2 LSY fatigue crack growth ...................... 75 1.3 Conclusion of SENT-type test .................... 84

Table 3 . 1 :

 31 LSY experimental loading conditionsFatigue crack growth results for experiments under constant-amplitude loading conditions for three different maximum strain values are presented in Table3.2; the tests with max-imum strain of 0.5% and 0.7% were doubled. For each test, a crack length 1 of about 10mm was obtained in a few thousands of cycles. The crack length here includes the notch size of 1.5mm and the pre-crack length. The crack growth under different loading conditions was plotted with the number of cycles in Fig.3.2.

	Specimen number	ε min	ε max	ε a
	1884-4A	0.06%	0.3%	0.12%
	1884-7A	0.1%	0.5%	0.2%
	1884-3A	0.1%	0.5%	0.2%
	1884-5A	0.14%	0.7%	0.28%
	1884-2A	0.14%	0.7%	0.28%
	1884-6A 1884-13A	0.14% → 0.1% 0.7% → 0.5% 0.28% → 0.2% 0.14% → 0.06% 0.7% → 0.3% 0.28% → 0.12%
	Figure 3.1: Illustration of variable-amplitude loading

Table 3 . 2 :

 32 Fatigue crack propagation experiment under constant-amplitude loading conditions

	Specimen ε max Number of cycles	Pre-crack (mm)	Crack length (mm)
	1884-4A 0.3%	4950	1.42	8.06
	1884-7A 0.5%	1800	1.12	9.84
	1884-3A 0.5%	1950	1.49	9.12
	1884-5A 0.7%	1320	1.84	11.65
	1884-2A 0.7%	1220	1.48	10.77

Table 3 . 8 :

 38 Crack length and crack growth rate on PACIFIC disc

	Crack		Pre-crack length a 0 (mm)	Total crack length a t (mm)	∆a = a 0 -a t (mm)	Mean da/dN (µm/cycle)
	Top	C1	8.6	14.5	5.9	10.2
		C2	2.9	4.5	1.4	2.8
		C3	7.9	13.6	5.5	9.8
		C4	4.7	8.2	3.5	5.6
	Bottom C1	10.4	15.5	5.1	8.1
		C2	0.6	2.25	1.65	2.5
		C3	8.4	13.9	5.5	9.2
		C4	2.1	7.0	4.9	8

  thus the evolution of K xc with crack length propagation could be obtained by the following equation: K m represents the size of monotonic elastic domain, in other words, it also corresponds to the external load which allows extending the monotonic plasticity in the crack tip region. Therefore we have:

			∂K xc ∂a	=	∂K xm ∂a	+	∂K m ∂a	-	∂K c ∂a	(4.38)
	3.3.2 Evolution of internal variables for monotonic behavior
	∂K m ∂a	and	∂K xm ∂a
	As discussed before, ρ∝K m	2	(4.39)

  Identification of the crack growth law ................150 2 Analysis of the PACIFIC test ........................159

	1 Identification and implementation of the incremental
	model for 316L SS
			Chapter 5
		Identification and validation of the
			incremental model
	Contents
	1	Identification and implementation of the incremental model for 316L SS 136
		1.1	Overview of the parameters in the incremental model ........136
		1.2	Identification of the parameters of the plastic blunting law for 316L
			SS ...................................136
		1.3	Implementation of the incremental model ..............144
		1.4

Crack growth modelling: Development of the incremental model

crack length = notch(1.5mm) + pre-crack + fatigue crack growth

transformation of the points on the circle from (rcos(θ), rsin(θ)) to (rcos(θ)+rg x (θ), rsin(θ)+rg y (θ))

Appendix A

Incremental variation

The internal variables K m , K xm , K c and K xc evolve with crack propagation (a) and plasticity (ρ). Their variations during one loading increment are calculated according different loading phases (cyclic plasticity or monotonic plasticity), noted by the intermediate variations δK a mono m , δK a mono xm , δK

, δK a mono c and δK a cycl c . Their evolutions in different domains ara calculated with the following equations.

Contact

When the crack faces are in contact, there is no crack propagation. Consequently, no evolution of variables takes place. However, in order to represent effect of negative loading ratio, K xc could evolve when crack is on contact.(cf. §1.3.3).

Elasticity

In elastic phase, ρ remains constant. Therefore, there is no evolution of variables if no temporal behaviour is exhibited (oxidation,...).

Cyclic plasticity

Calculation of δK ρ c

The size of the cyclic elastic domain 2K c depends on ρ mono with 2K c = a √ ρ mono + b.

Then we have the evolution of K ρ during ∆K I without crack growth expressed as follow:

Calculation of δK a c

The K c decreases with the crack growth. An analogue expression as ∂K m ∂a is proposed for K c against a, namely, ∂K c ∂a = -p a K c . Therefore, the variation of K c with crack growth in