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Fatigue in nuclear components: industrial context of the study
Operating principle of a nuclear plant
There are different nuclear reactor technologies. In France, the most widespread technology is the pressurized water reactor (PWR). Research has been underway for several years to develop the ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration) reactor, which is an industrial demonstrator of a sodium-cooled fast neutron reactor (FNR-Na). Fig. 1-2 and Fig. 1-3 show schematic diagrams respectively of a PWR and a FNR-Na. Whether the reactor is pressurized water or fast neutron, the basic principle is the same. The fission of a nuclear fuel (Uranium 235 in the case of a PWR, or Uranium 238+Plutonium mixture in the case of FNR-Na/He) produces heat. This heat is transported by a heat transfer fluid (water in the case of PWRs and Na/He in the case of FNRNa/ He) in a closed circuit (primary circuit) and transmitted via a heat exchanger to a closed secondary circuit (secondary circuit) in which liquid Na/He gas (in the case of FNR-Na/He), or water (in the case of PWRs) circulates. In the case of PWRs, the water in the secondary circuit is transformed into steam, which during its expansion activates a turbine coupled to a generator that produces electricity. In the case of FNR-Na, the sodium circulating in the secondary circuit transmits its heat to the tertiary circuit (where the sodium/gas conversion takes place) via a sodium/gas heat exchanger (SGHE). As before, the gas activates the turbine, which produces electricity. At the end of the chain, there is a cooling circuit in which water and/or air circulates to cool the steam leaving the turbine and transform it into water (via a condenser) and returned to the steam generator.
Fatigue of nuclear components
Reactor start-up and shut-down cycles are a source of cyclic loading that can lead to fatigue damage of individual components. The circulation of fluids at different temperatures in the piping system also causes fatigue damage. Various incidents have already been reported in nuclear power plants in different com ponents: manifold, steam generator, primary pump, piping, valves, turbines,… [Osterstock, 2013]. The Civaux incident is at the origin of many studies at CEA, EDF, IRSN and different universities. It also motivates this work and occurred on May 12, 1998 at the Civaux power plant. A leak of 30 m3/h was detected in the reactor cooling system when the reactor was shut-down, after only 1500 hours of operation. From then on, metallurgical and mechanical studies have been carried out to determine the cause of the leak as well as thermal hydraulics and experimental ones. Expertise operations reveal a traversing crack of length 180 mm long in the outer skin and 250 mmlong in the inner skin along a longitudinalweld in an AISI 304L austenitic stainless steel bend (Fig. 1-4, [Maillot, 2003]). Thermal crack networks were also observed in the vicinity of the mixing tee. In fact, the turbulences induced by the mixing of hot and cold fluids cause temperature variations of up to 150 ∘C in the internal part of the pipe walls, as can be observed in Fig. 1-5 obtained by thermohydraulic simulation. A temperature gradient is established in the thickness of the pipe. The hot zones deform independently of the cold zones inducing deformation incompatibilities. Strain compatibility is restored by the creation of elastic strains and then internal stresses (Fig. 1-6).
These thermally-induced stresses, which vary over time in the same way as temperature varies, characterizes a complex fatigue loading. Following the Civaux incident, all bends in the French nuclear fleet were replaced by components accounting for geometric modifications to limit turbulence in the mix. In spite of this, fatigue cracks were still detected recently on pipes in the secondary circuit of some PWRs.
In addition, in the FNR-Na technology, a Sodium/Gas Heat Exchanger (SGHE) is planned. This heat exchanger, consisting of several modules (Fig. 1- 7) inside which liquid sodium and nitrogen under pressure circulate, is made of 316L(N) austenitic stainless steel using the hot isostatic pressing process of millimeter thick plates. Due to the manufacturing process, grain size up to a few hundred microns are achieved for component thicknesses of around 1 mm. This induces specific problems with regard to the fatigue strength of these SGHE components and, above all, with regard to their design. In fact, space separation is no more guarantied in such structure.
It is true that the discussion relates to the thermomechanical fatigue. But our aim is to treat the initiation of fatigue cracks by reducing to mechanical loading at constant temperature.
As we have just seen, fatigue is a phenomenon present in all areas of industry (aeronautics, railways, automotive, energy, manufacturing, etc.).
More research in this field especially at low scale may make it possible to better control this damage mechanism by understanding the fundamental mechanisms of fatigue crack initiation and propagation. The effect of crystal orientation, the influence of temperature and stacking fault energy and the effect of loading mode are the questions to which we will attempt to provide partial answers through our present contribution.
In chapter 2, we present the state of the art on the initiation and propagation established in the wall of thickness (i). If the hot and cold zones expand independently, this leads to a strain incompatibility (ii) which are restored by the creation of elastic strains and then internal stresses (iii) [Déprés, 2004] [Cachon et al., 2015] and (b) sectional view of a module of the exchanger [Hazan, 2019] of fatigue cracks in metals and alloys and we note the shortcomings of the state of the art in fatigue modelling and lack of consensus regarding fatigue mechanisms while indicating the outlines of this manuscript.
State of the art
Ceux qui sont férus de pratique sans posséder la science sont comme le pilote qui s’embarque sans timon ni boussole et ne saurait jamais avec certitude où il va.
Abstract
A short review of fatigue of materials is provided in the following lines.
First, a review of the fundamental concepts of fracture mechanics and fatigue of materials is presented: the stress concentration, stress singularity, the stress intensity factor, loading modes, fatigue life approaches (S-N curves, strain-life approach, fracture mechanics approach). Secondly, a literature review on extrusion growth and fatigue cracks initiation sites (persistent slip bands, twin boundaries, grain boundaries) and mechanisms is displayed followed by a review on fatigue crack propagation in particular fatigue crack growth mechanisms (blunting-resharpening, void coalescence) and the effect of environment.
Fatigue of metallic materials: background information and theory
Fundamental concepts of Fracture Mechanics
Fracture mechanics is concerned by stress concentrators and fracture modes.
The fracture can occur from a pre-existing flaw. Considering a plate pierced with a hole of elliptical section, the Fracture Mechanics shows that with such a defect, the stress distributions are inhomogeneous and lead to stress concentrations in the vicinity of the flaw. The stress concentration factor 𝐾𝑡 is then defined as the ratio of the maximum stress reached in the vicinity of the flaw and the nominal stress applied to the plate.
𝐾𝑡 = 𝜎𝑚𝑎𝑥 𝜎𝑛𝑜𝑚 (2.1)
This parameter therefore provides an idea of the local stresses prevailing in the vicinity of a flaw with regard to the macroscopic applied stresses. The value of the stress concentration coefficient varies according to the geometry of the defect.
In the case of an elliptical defect with major axis 2𝑎 and minor axis 2𝑏, the stress concentration coefficient is given by: 𝐾𝑡 = 1 + 2
By assimilating a perfect crack to a infinitely flattened elliptical defect (𝑏 ≪ 𝑎), one notices that 𝐾𝑡 tends towards infinity and therefore such approach does not allow to characterize the local fields in the vicinity of the defect. The introduction of the stress intensity factor (SIF) by Irwin [Irwin, 1957] following the investigations of Westergaard [Westergaard, 1939] made possible the characterization of the distribution of mechanical fields in the vicinity of a crack within the framework of linear elastic fracture mechanics (LEFM). Indeed, with the asymptotic development of the stress field in the vicinity of the crack one can write: 𝜎𝑖𝑗 (𝑟, 𝜃) = 𝐾 √ 2𝜋𝑟 𝑓𝑖𝑗 (𝜃) (2.3)
where 𝐾 is the SIF, 𝑟 the position of a pointM(Fig. 2-1) with respect to the crack tip and 𝑓𝑖𝑗 (𝜃) an angular distribution function which depends on the loading mode.
The different loading modes are (see Fig. 2-2):
• Mode I or opening mode, characterized by an opening of crack lips in the normal direction to the crack plane.
• Mode II or shear mode characterized by in-plane shearing of crack lips, in the direction normal to the crack front lips in the direction parallel to the crack front.
Another way to characterize the stress singularity near crack tip is the use of contour integrals including the Rice integral. The latter is defined by: 𝑤𝑑𝑦 − 𝑇 𝜕𝑢 𝜕𝑥 )︂ 𝑑𝑆 (2.4) with an integration contour surrounding the end of the crack (Fig. 2-3), 𝑇 and 𝑢 denotes the stress and displacement vectors in a point of . 𝑤 is the strain energy density. The J-integral is path-independent in plastic materials provided that there is no non-proportional loading. Considering that the J-integral allows the characterization of the singularity of stresses at crack-tip as well as the stress intensity factor, it is shown that in LEFM framework, there exists a relation between the integral J and the stress intensity factor. For an isotropic material, one can write: 𝐽 = 𝐾2 𝐼 𝐸* (2.5) with 𝐸* = 𝐸 under plane stress conditions and 𝐸* = 𝐸/(1 − 𝜈2) for plane strain condition. The Young’s modulus is denoted by 𝐸 and 𝜈 denotes the Poisson ratio.
Fundamentals of fatigue of materials
Fatigue loading
There are different fatigue loading paths. In this thesis, we will limit ourselves to constant amplitude loading. Various parameters allow the characterization of this type of loading namely:
• The form of the loading: For a sinusoidal loading with constant amplitude, the applied stress obeys the following mathematical formulation: 𝜎𝑖𝑗 (𝑡) = 𝜎𝑎,𝑖𝑗 sin (𝜔𝑡) + 𝜎𝑚,𝑖𝑗 (2.6) where 𝜎𝑎 is the amplitude of the stress and 𝜎𝑚 is the mean stress. 𝑓 = 𝜔/2𝜋 is the loading frequency. The amplitude of the applied stress and the mean stress are calculated respectively by Equation 2.7 and Equation 2.8: 𝜎𝑎,𝑖𝑗 = [max (𝜎𝑖𝑗 (𝑡)) − min (𝜎𝑖𝑗 (𝑡))] (2.7) 𝜎𝑚,𝑖𝑗 = [max (𝜎𝑖𝑗 (𝑡)) + min (𝜎𝑖𝑗 (𝑡))] (2.8)
• Load ratio: The stress ratio can be defined as: 𝑅𝜎 = 𝜎𝑚𝑖𝑛 𝜎𝑚𝑎𝑥 (2.9)
According to the value of the loading ratio, one distinguishes the alternating loading (for which the load ratio is equal to −1), the repeated loading (for which the load ratio is 0). Another type of load ratio conventionally used for fracture analysis of cracked bodies (and also used in this work) is the one based on stress intensity factors and defined by: 𝑅𝐾 = 𝐾𝑚𝑖𝑛 𝐾𝑚𝑎𝑥 (2.10)
During cyclic loading, the properties of metallic materials are modified. Microdamage accumulates may becomes fatal. In practice, the fatigue of materials is approached in two different ways: the recording of the Wöhler curves (S-N curves) and the measurement of the crack growth rates.
Fatigue life approaches
S-N Curves The S-N curve or Wöhler curve is a curve that allows the evaluation of the lifetime of a material (in terms of the number of cycles to failure) as a function of the magnitude of the applied stresses. It is also called the endurance curve. Fig. 2-4 shows an example of the Wöhler curve. An increase in the lifetime when the magnitude of the applied stresses decreases is observed. We can split this curve into three zones:
• Low cycle fatigue zone (zone 1) for which the applied stresses are higher than the yield stress of the material. This zone is also called the oligocyclic plastic fatigue zone. In this area the fracture occurs after a small number of cycles (few cycles to approximately 104 cycles)
• Zone of limited endurance (zone 2) for which failure occurs after a large number of cycles but limited (fatigue life between 105 and 106 cycles)
• Zone of fatigue limit or safety zone (zone 3) which corresponds to low stress loads. In this zone, it is estimated that the fracture cannot occur before a higher number of cycles: 106- 107 cycles. However, gigacyclic fatigue tests question the existence of this limit ([Mughrabi, 1999, Mughrabi, 2006, Stanzl-Tschegg et al., 2007]) Strain-life approach In addition to S-N curves, strain-life approaches (Fig. 2-5) are also often used to estimated fatigue crack initiation.
Table of contents :
1 Context and motivation
1.1 Introduction
1.2 Fatigue in nuclear components: industrial context of the study
2 State of the art
2.1 Fatigue of metallic materials: background information and theory
2.2 Literature review on fatigue crack initiation
2.3 Literature review on fatigue crack propagation
2.4 Conclusion
3 Growth of extrusions in cyclically deformed f.c.c. metals and alloys
3.1 Introduction
3.2 Constitutive equations
3.3 Modelling hypotheses
3.4 Prediction of extrusion growth
3.5 Stress field analysis and fatigue crack initiation
3.6 Conclusion
4 Plastic deformation and underlying crystal defects in the vicinity of fatigue crack-tip
4.1 Introduction
4.2 Computational method
4.3 Dislocation emission from crack-tip and plastic deformation localization
4.4 Plastic deformation mechanisms and lattice defects
4.5 Plastic irreversibility
4.6 Conclusion
5 Fatigue crack growth in copper single crystals at room temperature
5.1 Introduction
5.2 Fatigue crack growth rates
5.3 Fatigue crack growth mechanisms at nanoscale
5.4 Crack geometry
5.5 Conclusion
6 Effect of temperature and stacking fault energy
6.1 Introduction
6.2 Computational methodology
6.3 Effect of temperature on fatigue crack behavior
6.4 Effect of stacking fault energy
6.5 Conclusion
7 Fatigue crack growth under mixed mode
7.1 Introduction
7.2 Computational methods
7.3 Plastic deformation
7.4 Propagation
7.5 Conclusion
8 Conclusions and future research directions
A Engineering moduli
B Metals properties
C LEFM stress fields around crack-tip in anisotropic bodies
D Proportion of different types of dislocations in copper single crystals
D.1 Effect of crystal orientation
D.2 Effect of temperature
D.3 Proportion of different dislocation types in different fcc metals Fatigue crack initiation and propagation
D.4 Effect of loading amplitude on the proportion of different dislocation types in nickel
D.5 Proportion of different dislocation type under mixed mode loading
E Crack propagation rates
E.1 Effect of temperature
E.2 Effect of SFE (material)
F Dislocation microstructures
F.1 Effect of temperature
Bibliography
