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Oxidation-assisted Cracking
15 Oxidation-assisted Cracking
15.1. Introduction Oxidation-assisted cracking (OAC) is a particular type of damaging process called “dynamic embrittlement”. Dynamic embrittlement is a fragile rupture process which implies the ingress, diffusion and segregation of embrittling elements at the grain boundaries leading to their debonding. These embrittling elements may be provided by the material itself, as in the case of sulfur in steels or tin in copper alloys, or by the material’s environment, notably oxygen’s partial pressure. One of the most-studied cases is OAC for nickel-based alloys for which the penetration and segregation of oxygen, often associated with intergranular oxidation, change the crack propagation mode from transgranular to intergranular and significantly increase the crack propagation rate [KRU 04, WOO 05, CHA 15]. In the aeronautical industry, as in the nuclear industry, this phenomenon is responsible for numerous cases of in-service cracking [HOC 94, CHA 97]. To study the propagation rate’s susceptibility to the chemical environment, numerous mechanical tests are conventionally performed in laboratory air or a vacuum. Figure 15.1 [PED 82] shows the propagation rates measured during low-frequency fatigue-creep tests (trapezoidal mechanical cycle: 10s–300s–10s) in the case of alloy 718 tested at 650°C in different environments. It is worth mentioning that oxidation has a significant effect on the crack propagation rate. In addition to the change in the propagation rate, the cracks propagate via the grain boundaries in air or the core of the grains in a vacuum. It should be noted that intergranular cracks might propagate at a maximum speed which may reach values of µm/s. The latter observation demonstrates the scientific Chapter written by Benoît TER-OVANESSIAN, Aurélien VILLANI, Éric ANDRIEU and Samuel FOREST.
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problems related to OAC. It is effectively necessary to take into account the oxidation modes, their kinetics, the local mechanics and the metallurgical state near the defect in order to propose physical modeling which is well underpinned.
Figure 15.1. Effect of oxidation and microstructure on the crack propagation rate in fatigue-creep test at 650°C for alloy 718
The time dependence for the OAC phenomenon is intrinsically linked to the temperature, the metallurgical state, the oxygen partial pressure, the oxide type and growth mode, the diffusivity of the weakening species within the grain boundary (increased or not by the local stress state) and finally the stress level (macroscopically and locally). Figure 15.2 illustrates an important point relative to the effect of oxygen’s partial pressure on the crack propagation rate at constant ΔK. It can be noted in Figure 15.2 that the propagation rate does not vary linearly with oxygen’s partial pressure and that a pressure threshold exists beyond which the propagation rate varies abruptly. This behavior is due to a change in the oxidation mode [GER 95].
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Figure 15.2. Evolution of the crack propagation rate at 650°C according to oxygen’s partial pressure at constant ΔK for different fatigue cycles
Mechanical loading plays multiple roles in the damage mechanism. It may act as a motor (tensile) or brake (compression) for the diffusion by acting on the flux and the solubility of the weakening species. In these interactions, the specific role of the mean stress or the deviatoric part of the stress field has to be considered. Mechanical loading may also act on the growth kinetics and the distribution of the growth stresses associated with oxide formation. Furthermore, when mechanical loading leads to local plastic or viscoplastic deformation, a segregation of impurities transported by the dislocations might lead to modification of the grain boundaries’ metallurgical state, and then to its sensitivity to intergranular oxidation. Reciprocally, in addition to the oxygen segregation at grain boundaries, their oxidation (the formation of intergranular oxide) plays an important role in the rupture. The geometry of the oxide tip allows us to test the grain boundaries toughness [AND 92]. The latter phenomenon may be considered as a geometrical assistance for the initiation and propagation of cracks. The relative contribution of these different factors to damage and their time dependence leads to various cases of OAC which are mostly differentiated by their cracking rates. When OAC occurs, a range of average propagation rates from a few mm/h to a few nm/h is covered [CHA 97]. Without being controlled systematically
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by a single mechanism, the different cases of OAC can be separated following a comparative analysis between the kinetics of fundamental mechanisms and the time scale associated with OAC rupture. Therefore, the objective of this chapter is not to focus our attention on one type of mechanism but rather to extend the subject by presenting different mechanisms, which are to various degrees potentially involved in OAC depending on the local conditions for mechanical and physicochemical loadings. 15.2. Multiprocess assistance: oxidation’s role Oxidation-assisted cracking’s complexity involving the contribution of chemical assistance, mechanical assistance and synergies makes it important to take into account the different mechanisms or sequences of mechanisms in order to understand the damage process, its kinetics and its mitigation. As an illustration, Figure 15.3 presents the evolution of the crack propagation rate during the superposition of a fatigue cycle and an oxidation cycle [MOL 97]. It is worth mentioning that the propagation rate only changes when the environmental cycle is superimposed to the ascending part of the mechanical cycle. This behavior demonstrates firstly the strong coupling between the local mechanical state and the damage induced by the oxide growth and secondly, the beneficial effect of unloading on the resistance to OAC.
Figure 15.3. Evolution of the crack propagation rate in low frequency fatigue according to the relative position of the environmental cycle with the mechanical loading cycle (alloy 718 at 650°C)
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Assuming that oxidation acts as the main chemical assistance, distinctions must be made to understand its consequences on the OAC phenomenon. Indeed, generally, oxidation of a metal at high or intermediate temperatures is classically described and studied according to three designations: external oxidation, internal oxidation and intergranular oxidation. The penetration at the grain boundary of “free” oxygen (oxygen in the solid solution) may be included in the latter process. Depending on the nature of the environment and the exposition temperature, these oxidation mechanisms are promoted to different extents and affect the material’s life via the mechanisms which are specific to them. 15.2.1. External oxidation As the external oxidation mechanisms have been detailed in a previous chapter, only some notions will be addressed here. The formation and the growth of an external oxide are due to an oxide-reduction reaction where the metal is oxidized and the oxygen is reduced. The formation of these oxides depends on the temperature, partial pressure in O2 and presence of water/steam or other aggressive gases. It is therefore mainly a thermodynamically controlled process. Two growth modes govern the thickening rate of the oxide: when the growth is controlled by the diffusion of cations, the oxide formation is located at the oxide-environment interface; when the growth is controlled by the diffusion of anions, the oxide formation is then located at the metal–oxide interface. These two modes may sometimes coexist. Nevertheless, in both cases, the metal–oxide interface mobility is a crucial issue. Indeed, the metal-oxide interface is displaced during oxidation in order to support the metal’s consumption (cationic growth) or to generate the space to accommodate the expansion in volume due to the oxide formation at the interface (anionic growth). This mobility must be included during the modeling of these phenomena. In these two situations, it is principally the flux of vacancies which will assure the mobility of the interface, provided the latter is not anchored by structural effects due to the geometry of the component or defect. For the case of cationic growth, when the interface is pinned, the injection of vacancies in the substrate leads to a supersaturation in vacancies. This supersaturation will lead to the formation of pores or cavities in the substrate, or even the recovery of the deformation structures by climbing dislocations in order to maintain the concentration of vacancies in the substrate at thermal equilibrium [PER 04a, PER 05]. Figure 15.4 illustrates the cavitation phenomenon induced by the injection of vacancies. For the case of anionic growth, the pinning of the external oxide layer restrains the setting of the flux of vacancies from the substrate towards the metal–oxide interface needed to generate the space for the oxide’s formation. In this case, vacancy sources activated within the metal are dislocations (climbing) and porosity when they preexist. These mechanisms, when they occur, create local
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relaxation of the stresses via the recovery of deformation structures. When they do not occur, the substrate is subjected to growth stresses, which may mechanically affect the geometry of the structural component. Nevertheless, these vacancies may contain oxygen and thereby not only lose their status as a thermal vacancy but also contribute to oxygen diffusion via substitutional sites. At the bottom of a grain boundary, these same mechanisms have specific consequences according to the grain boundary’s geometry and its disorientation. In the latter case, the contribution of external oxidation to OAC based on oxygen enrichment of the grain boundary will initiate intergranular oxidation.
Figure 15.4. Formation of an intergranular cavity in a strip of nickel oxidized on the two faces under laboratory air at 1000°C during 48h [PER 04b]
Furthermore, preferential oxidation of certain alloying elements has the consequence of generating a zone affected by oxidation in the substrate within which the chemical composition of the material is significantly modified. Thereby, the local metallurgical state is modified in terms of the precipitation state and the matrix lattice parameter. The consequences of preferential oxidation on the mechanical behavior of the affected zone may also slow down OAC by relaxing the local stresses or blunting the crack tip. 15.2.2. Internal oxidation In some cases, external oxidation is accompanied by internal oxidation. The preferential oxidation of certain elements in the matrix is indicative of penetration and diffusion of oxygen within the metallic material [SCO 99]. Beyond the “marker” effect by the precipitation of oxides, internal oxidation will also induce a
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change in the substrate’s chemical composition. Furthermore, like the concerns raised for external oxidation, the expansion in volume due to oxide formation may be accommodated by viscoplastic deformation or by condensation of vacancies coming from various sources.
Figure 15.5. Internal oxidation of an Fe-Ni (60%at )-Cr(7.5% at) alloy exposed for 100h at 1150°C in a RHINES pack [PRI 17]
Figure 15.5 [PRI 17] illustrates the latter phenomenon well. The internal oxide formation induces the formation of metallic protuberances at the sample’s surface. In this work, the oxygen’s partial pressure corresponding to the oxide’s dissociation does not permit the development of an external oxide layer. Consequently, the source of vacancies located at the sample’s surface and the flux of vacancies generates an opposite flux of iron and nickel atoms. The consequences of these phenomena on OAC are many. For example, the precipitation of oxides is hardening but the flux of vacancies associated with the formation of oxides are instead softening. It is therefore the local behavior of the zone affected by oxidation, resulting from the contribution of antagonistic phenomena, which will affect OAC.
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15.2.3. Intergranular oxidation Although the fundamental mechanisms involved are identical to those involved in so-called “external” oxidation (anionic or cationic growth, etc.), germination kinetics such as oxide growth at the grain boundaries are different from the behavior of the bulk material, as often observed with cross-sectional analysis. Indeed, local conditions are slightly different. In this case, the oxidation front grows at an interface, which may not have in its initial state the same chemical composition as the bulk material. Indeed, a grain boundary has a specific thermo-mechanical history resulting from the different metallurgical transformations, which monitor its chemical composition and its degree of disorientation. At the “substrate’s” scale, it no longer has the same crystallographic structure as the grain itself, which gives rise to diffusion coefficients that are significantly different with regards to the grain bulk. Grain boundaries generally react as short diffusion circuits. The consequences of intergranular oxidation are similar to external oxidation while being more damaging because they are at a more local scale. The critical level of the damage mainly depends on the grain boundary geometry and the depth of oxide penetration. The cohesion of this grain–grain interface can be affected in different ways: reduction of cohesion energy by oxygen segregation, edge effect due to oxide formation, selective pumping of an element involved in the formation of the external oxide (chromium depletion for example), formation of pores or cavities, etc. Even if these different items are present at the interface, intergranular crack will only initiate beyond a critical stress which, to be reached, requires a dedicated local mechanical loading. 15.3. From multiple processes to coupled processes Oxidation generates chemical, mechanical and microstructural modification of metallic materials at the core of the grain as well as at the grain boundary. The various consequences of this oxidation may act as brakes or motors to OAC crack initiation and propagation. However, as mentioned previously, grain boundary rupture can occur only if a critical stress is reached locally. The coupling between the chemical process and mechanical loading is therefore necessary for OAC cracking. The study of these synergies is then required in order to understand OAC. 15.3.1. Mechanical-diffusion coupling In the previous paragraphs, different mechanisms influencing OAC have been detailed. Not only various and numerous, these processes are often also interdependent. The importance of species diffusion has been highlighted in the description of the mechanisms. However, the mechanical state at the crack tip or
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generated by a macroscopically mechanical loading may interfere with it. To better understand the mechanics-diffusion coupling, its modeling is therefore essential. A well-known example is Cottrell’s atmosphere [COT 75]: under its own stress field, a dislocation locally reorganizes the distribution of vacancies (Figure 15.6) and solutes of a material. Although known for a long time, this phenomenon is still the object of studies, whether at the atomic scale [MIS 16], or at the continuum scale [CAH 13].
Figure 15.6. Vacancy concentration field around an edge dislocation in an infinite medium, given by a continuum theory, and normalized by the homogeneous concentration at thermal equilibrium. Since the analytical solution of the stress field is singular, the dislocation core has been omitted. For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
In this paragraph, the numerical approach considers the continuum scale. Indeed, the conventional thermodynamic framework for continuum mechanics is perfectly adapted to the simulation of mechanical-diffusion coupling. The key idea here is to introduce into the strain components an eigenstrain depending on the local concentration fields. For example, in the case of only one chemical species, the eigenstrain is expressed as: 𝜀∗ 𝑐 = 𝜂 𝑐 − 𝑐
[15.1]
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where c is the local concentration of the species considered, 𝑐 is its concentration at thermal equilibrium and η the coupling factor. In general, η is a tensor proportional to the identity, or diagonal. As for a thermo-mechanical coupling, a local variation of concentration leads to a local volume change. Indeed, under- or over-saturation of alloying elements, or of vacancies, locally disturbs the crystal lattice. This coupling term having been introduced, conventional application of thermodynamics for continuum mechanics leads directly to an expression of the flux vector of the type: 𝐽 = −𝐷𝛻𝑐 + 𝑓 𝜂 𝛻 𝑡𝑟𝑎𝑐𝑒 𝜎
[15.2]
which is nothing less than a modified Fick’s law, with D the diffusion coefficient. Note that the second term corresponds to the contribution of the mechanics in the diffusion potential, and is proportional to the coupling term η, as well as to the gradient of the trace of the stress tensor. It is noted that the general theory also gives the concentration field at equilibrium, according to the boundary conditions, but the reader is referred to [VIL 14] for more information. The influence of mechanics on the diffusion is therefore obvious. In principle, the theory mentioned previously also gives the influence of the concentration field on the stress field at equilibrium, via the use of elastic constants derived at constant chemical potential. In practice, with typically low concentrations, this effect is negligible. To use the example of the dislocation, the ratio of the hydrostatic pressure fields given by a coupled and uncoupled elastic theory is typically 0.999. Except for particular cases, irradiation for example, it is possible to use conventional elastic constants in a coupled problem. The equations for the model briefly introduced above can be solved relatively simply by finite elements method [VIL 14]. In the following example, a hollow and notched cylindrical test specimen is filled with hydrogen at constant partial pressure, Figure 15.7(a). In the absence of mechanical loading, hydrogen diffuses until the concentration is homogeneous in the overall specimen. However, if the test specimen is loaded with tensile or compressive stress along the rotating axis, the notch will play the role of a stress concentration. Following equation [15.2], the concentration at equilibrium will be therefore heterogeneous. For tensile tests, it is in the bottom of the notch where the concentration is the highest, with a value about 1.05 times higher than elsewhere within the test specimen for this example (Figure 15.7(b)). In the case of a crack, the mechanical solution in elasticity is singular, just like for the dislocation given as an example at the start of this section. Even if the use of
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a viscoplastic model removes the singularity, the crack tip’s mechanical field remains strongly nonlinear, as we will see in section 15.3.3. Therefore, the local concentration at the crack tip can be strongly modified by the crack’s stress field, and thus significantly impacts OAC kinetics.
Figure 15.7. An interstitial concentration of hydrogen is imposed on the internal face of a notched hollow cylindrical specimen, modeled in a). The specimen is loaded mechanically along its rotating axis, and the concentration at equilibrium in the section of the specimen is presented in b) for tensile loading, and in c) for compressive loading. For a color version of this figure, see www.iste.co.uk/blanc/ coupling.zip
15.3.2. Coupling intergranular mechanical-diffusion-oxidation behavior As previously shown, the diffusion of the solutes is modified when a stress field is applied, highlighting the mechanical-diffusion coupling which may be involved in OAC. Observed macroscopically, this contribution certainly locally influences the grain boundary (where the diffusivity of the species is already different) or the crack tip. However, the chemical species’ mobility in a material under stress can also be modified by plastic deformation, or plastic flow. Solute-dislocation interactions, whether they are viscous drag type or pinned/released type, are known to be responsible for irregularities in mechanical behavior such as dynamic strain ageing, Luders or Portevin-Le Chatelier (PLC) effects. Thermally activated, these different deformation modes also depend on the strain rate and metallurgical parameters,
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which jointly influence the solutes’ diffusivity and the mobility and localization of dislocations. When these deformation modes are observed, the associated interactions play a role in the OAC process. Numerous works have been performed in the last few years to study the interaction between deformation modes and OAC sensitivity for nickel-based superalloys [FOU 00, FOU 01, GAR 08, DEL 07]. Their first observation is that during tensile tests in air for temperatures between 300°C and 700°C, the fracture surface of the alloy 718 presents no fragile intergranular zone when plastic instabilities are observed on the tensile curves. Conversely, the absence of these instabilities is directly linked to the emergence of fragile intergranular zones on the fracture surfaces, in the same temperature domain. These tests have also been performed in an inert atmosphere, and do not lead to fragile rupture. There is therefore a robust correlation between the OAC’s occurrence and the local plastic deformation modes. Figure 15.8 shows a temperature–strain rate map for alloy 718. The intergranular rupture–transgranular rupture frontier coincides with the dynamic strain ageing–PLC frontier. Different mechanical or chemical scenarios can be proposed to explain this damage mode. The PLC flow instabilities may be considered differently: propagation of a deformation band, limited germination and propagation of several bands and random germination of small bands. The amplitude of the instabilities on the tensile curve allows us in general to determine the type of the PLC. In general, this behavior is a result of the strain’s localization at the scale of the test specimen, which leads to a reduction of the material’s elongation in the case of rupture in an inert atmosphere. The deformation in a strip is very rapid and rather homogeneous. Consequently, the local strain rate is not very compatible with oxidation kinetics. Furthermore, a decrease in the stress is observed using a hard tensile testing machine (imposed displacement), making it less likely that defects will be opened which were not open before. This type of scenario applied in the plastic zone of a propagating crack tip leads to very rapid relaxation of the stress field at the tip of the defect and therefore inhibits OAC. It is also possible to imagine that this flow instability effect applied to a defect could inhibit crack initiation. However, whilst the loading mode is associated with dynamic strain ageing, the rupture becomes intergranular in an oxidizing atmosphere but remains transgranular in an inert atmosphere [MAX 16]. In this case, plastic strain is supposed to be distributed on the entire gage length of the specimen at the macroscopic scale. However, at the polycrystal scale, we believe that deformation heterogeneities generate significant intergranular stresses on certain grain–grain interfaces. Additional studies have shown that for these modes of plastic flow, it was necessary to locally associate a critical cumulated strain allowing the cohesive rupture of the grain boundary (Figure 15.8) [TER 12].
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Figure 15.8. Experimental mapping of the deformation modes and the rupture modes for alloy 718 in the temperature field 550–700°C. The cumulated plastic strain necessary for intergranular cracking in air is reported (see [TER 11]). For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
A more chemical approach establishes the relationship between the transport of solutes to the grain boundaries by dislocations, deformation modes and intergranular rupture. In the dynamic strain ageing regime, without any occurrence of PLC effects, the viscous drag of the solutes towards the grain boundaries and the zones with a high stress concentration can locally modify the chemistry of the grain boundary, already affected by the penetration of oxygen or oxide, and induce decohesion of the latter. In the case of a PLC effect, the pinning/unpinning mechanism does not lead to such grain boundary enrichment and local interactions with the dissolved oxygen. This chemical approach does not contradict the mechanical approach. The notion of damage accumulation can also be seen as an accumulation of solutes at the grain boundaries. The chemical species suspected to weaken the grain boundaries are the interstitial elements C, H, N, which react easily with oxygen. However, a recent study dedicated to this alloy, using internal friction measurements, showed that molybdenum was the alloying element most likely to induce plastic instabilities in the temperature range explored [MAX 16, MAX 18]. This work
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clearly highlights the coupling between mechanical behavior, chemical species’ mobility, intergranular oxidation and OAC sensitivity. The assumptions made in order to explain this type of damage warrant confirmation via the acquisition of new experimental or numerical data. 15.3.3. Simulation methods for mechanical-oxidation intergranular coupling The fascinating microstructure in Figure 15.9 contains all the features necessary for modeling oxidation and plasticity-assisted cracking. We see that the oxidation takes some preferred routes which will be possible locations for crack initiation: the intense plastic slip bands and the grain boundaries. The V shape of the oxidation tip at the slip bands shows the importance of oxygen’s accelerated diffusion in these zones and the angle of the tip providing information on the associated interface energy. Diffusion, oxidation, elasticity, viscoplasticity and damage are the mechanisms to couple in a unified modeling approach. In a global approach, multiple chemical reactions, which are linked to complex oxides and the electrochemical aspects, should also be added.
Figure 15.9. SEM microstructure for a 316L steel loaded in torsion during a fatigue-creep test and then exposed for one hour for oxidation, according to [WEI 92]. See also [WEI 93]
The phase field method is a thermodynamic framework for modeling the evolution of the phase change fronts and their interaction with diffusion and mechanics [STE 09, FIN 10]. To illustrate the potential as well as the complexity of this approach, we detail the example of chromium oxide formation in a
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(quansi-)stainless steel 316L. The model’s chemical part is described by the density of Helmholtz free energy which depends on the species concentrations (to simplify, O, Cr and Fe) and by the dissipation potential associated with the diffusion of these elements. The phase field is a variable passing smoothly from 0 in steel to 1 in the oxide. Its evolution is governed by the Allen–Cahn equation (also called Ginzburg–Landau), which includes a diffusive term and chemical driving forces for the front’s progress, as well as a mobility parameter for the interface. The internal oxidation front is treated as a diffuse interface whose thickness is regulated by the associated volumetric free energy and an equivalency with the interface energy. It is a characteristic of the phase field method in relation to the treatment of perfect interfaces for which changes in the shape, intersections and connections are difficult to handle. Intense slip bands and grain boundaries are diffusion short-circuits for oxygen and remain so once preferential oxidation paths have been established. This fact is taken into account in the simulation by the use of amplified diffusion coefficients in these zones. Figure 15.10 illustrates the formation of an intergranular oxide tip between two grains as well as the resulting depletion in chromium in the underlayer matrix. These concentration fields can be directly compared to experimental fields measured in SEM or TEM for example. The shape of the tip is not imposed; it results from the competition between interface energy, diffusion and also mechanics due to stresses generated by the eigenstrain associated with the oxide growth. This last parameter is essential for coupling the chemical thermodynamics with the mechanics but it is still badly known. We cannot be satisfied with the Pilling–Bedworth relationship, which takes into account the change in volume during the passage from the metallic phase to the oxide phase, because it leads to the calculation of stresses which are much higher than the residual stresses measured in the oxides [HUN 02]. The complexity of the process for oxide formation which results from the accommodation of the coherence stresses instead leads to an anisotropic eigenstrain, as proposed by Parise et al. [PAR 98], including components identified by an inverse approach starting from the stress measurements. The identification of the anisotropic eigenstrains via phenomenological or physical procedures is a major issue for model development in this field. The stresses generated around the oxidation tip are sufficiently high to trigger the activation of slip systems in the grains concerned. The theory and the numerical practice of crystal viscoplasticity are applicable in this case, as shown in Figure 15.11 (shown on the right), which gives the accumulated plastic slip field in the growth zone by assuming an ideal orientation of the FCC grain. Figure 15.11 (shown on the left) shows furthermore that hydrostatic stress is rather high which prompts us to utilize damage or local rupture criteria to predict crack initiation and propagation. Because of the tip’s complex geometry, we observe a tensile zone in front of the oxidation tip as well as at the back, even in the absence of external
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loading. The model then assumes a cracking process at the front contributing to intergranular damage, as well as at the back promoting the accelerated penetration of oxygen, or other aggressive species. The simulation of generalized intergranular damage is part of the toolbox of numerical simulations for crystalline plasticity coupled with diffusion, according to the work of Diard et al. [DIA 02] and Musienko and Cailletaud [MUS 09].
Figure 15.10. Chromium depletion during the oxidation of a grain boundary in a 316L steel [DER 15]. For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
Figure 15.11. Hydrostatic pressure field in the oxide and the substrate (to the left, the white corresponds to tensile values); cumulated plastic strain field in the substrate, the red exceeding one percent of plastic strain [DER 15]. For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
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15.4. Conclusion Although these couplings or synergies are observed experimentally and analyzed at increasingly fine scales, their modeling by numerical methods, which may rely on experimental validation, are an indispensable contribution to the understanding and discrimination of OAC mechanisms. This contribution provides us with the opportunity to firstly simulate and correctly predict the lifespan of structural elements and secondly, to find new ways of developing and/or designing in order to secure the structures exposed to this type of damage. The approach based on the phase field method, allowing us to couple the progress of an oxidation front with the diffusion of chemical elements, crystalline plasticity and damage, is extremely promising, even if the results remain qualitative nowadays. Considerable work is necessary to gain realism from the viewpoint of the coupling mechanisms involving microstructure, mechanical state and oxidation. 15.5. References [AND 92] ANDRIEU E., MOLINS R., GHONEM H. et al., “Intergranular crack tip oxidation mechanism in a nickel-based superalloy”, Material Science and Engineering A, vol. 154, pp. 21–28, 1992. [CAH 13] CAHN J., “Thermodynamic aspects of Cottrell atmospheres”, Philosophical Magazine, vol. 93, pp. 3741–3746, 2013. [CHA 97] CHASSAIGNE J.C., Fissuration à haute température du superalliage base nickel N18 élaboré par métallurgie des poudres, Etude du couplage mécanique environnement en point de fissure, PhD thesis, Ecole des Mines de Paris, 1997. [CHA 15] CHAN K.S., “A grain boundary fracture model for predicting dynamic embrittlement and oxidation-induced cracking in superalloys”, Metallurgical and Materials Transactions A, vol. 46, pp. 2491–2505, 2015. [COT 75] COTTRELL A., An Introduction to Metallurgy, 2nd ed., Edward Arnold, London, 1975. [DEL 07] DELEUME J., Facteurs métallurgiques et mécaniques contrôlant l’amorçage de défauts de corrosion sous contrainte dans l’alliage 718 en milieu primaire des réacteurs à eau sous pression, PhD thesis, INP Toulouse, 2007. [DER 15] DE RANCOURT V., Modelling the oxidation of polycrystalline austenitic stainless steels using a phase field approach coupled with mechanics, PhD thesis, Mines ParisTech, 2015.
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[DER 16] DE RANCOURT V., APPOLAIRE B., FOREST S. et al., “Homogenization of viscoplastic constitutive laws within a phase field approach”, Journal of the Mechanics and Physics of Solids, vol. 88, pp. 291–319, 2016. [DIA 02] DIARD O., LECLERQ S., ROUSSELIER G. et al., “Distribution of normal stress at grain boundaries in multicrystals: Application to an intergranular damage modeling”, Computational Materials Science, vol. 25, pp. 73–84, 2002. [FIN 10] FINEL A., LE BOUAR Y., GAUBERT A. et al., “Phase field methods: Microstructures, mechanical properties and complexity”, Comptes Rendus Physique, vol. 11, pp. 245–256, 2010. [FOU 00] FOURNIER L., Interactions corrosion-déformation dans l’alliage 718 : Application à la corrosion sous contrainte en milieu aqueux supercritique et recherché d’une solution matériau pour le procédé d’oxydation hydrothermale, PhD thesis, École des Mines de Saint Etienne, 2000. [FOU 01] FOURNIER L., DELAFOSSE D., MAGNIN T., “Oxidation induced intergranular cracking and Portevin-Le Chatelier effect in nickel base superalloy 718”, Material Science and Engineering A, vol. 316, pp. 166–173, 2001. [GAR 08] GARAT V., CLOUE J.M., POQUILLON D. et al., “Influence of Portevin-Le Chatelier effect on the rupture mode of alloy 718 specimens”, Journal of Nuclear Materials, vol. 375, pp. 95–101, 2008. [GER 95] GERVAIS-MOLINS R., Oxydation de superalliages à base de nickel. Identification des mécanismes et conséquences sur le mode de propagation des fissures en fatigue à haute température, PhD thesis, Ecole des Mines de Paris, 1995. [HOC 94] HOCHSTETTER G., Propagation des fissures à haute température dans le superalliage N 18 pour disques de turbomachine. Interactions entre la nature des sollicitations mécaniques et les effets d’oxydation, PhD thesis, Ecole des Mines de Paris, 1994. [HUN 02] HUNTZ A., CALVARIN AMIRI G. et al., “Comparison of oxidation-growth stresses in NiO film measured by deflection and calculated using creep analysis or finite-element modelling”, Oxidation of Metals, vol. 57, nos 5–6, pp. 499–521, 2002. [KRU 04] KRUPP U., MCMAHON JR C.J., “Dynamic embrittlement-time-dependent brittle fracture”, Journal of Alloys and Compounds, vol. 378, pp. 79–84, 2004. [MAX 14] MAX B., Comportement mécanique et couplage mécanique-oxydation dans l’alliage 718 : Effet des éléments en solution solide, PhD thesis, INP Toulouse, 2014. [MAX 18] MAX B., SAN JUAN J., NO M.L. et al., “Atomic species associated with the Portevin–Le Chatelier effect in superalloy 718 studied by mechanical spectroscopy”, Metallurgical and Materials Transactions A, vol. 46, pp. 1–12, 2018.
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[MIS 16] MISHIN Y., CAHN J., “Thermodynamics of Cottrell atmospheres tested by atomistic simulations”, Acta Materialia, vol. 117, pp. 197–206, 2016. [MOL 97] MOLINS R., HOCHSTETTER G., CHASSAIGNE J.C. et al., “Oxidation effects on the fatigue crack growth behaviour of alloy 718 at high temperature”, Acta Metallurgica Materiala, vol. 45, pp. 663–674, 1997. [MUS 09] MUSIENKO A., CAILLETAUD G., “Simulation of inter- and transgranular crack propagation in polycrystalline aggregates due to stress corrosion cracking”, Acta Materialia, vol. 57, pp. 3840–3855, 2009. [PAR 98] PARISE M., SICARDY O., CAILLETAUD G., “Modelling of the mechanical behavior of the metal–oxide system during Zr alloy oxidation”, Journal of Nuclear Materials, vol. 256, pp. 35–46, 1998. [PED 82] PEDRON J.P., PINEAU A., “The effect of microstructure and environment on the crack growth behavior of Inconel 718 at 650°C under fatigue, creep and combined loading”, Material Science and Engineering, vol. 56, pp. 143–156, 1982. [PER 04a] PERUSIN S., Etude des couplages oxydation et comportement mécanique dans différents matériaux modèles, PhD thesis, INP Toulouse, 2004. [PER 04b] PERUSIN S., VIGUIER B., MONCEAU D. et al., “Injection of vacancies at metal grain boundaries during the oxidation of nickel”, Acta Materialia, vol. 52, pp. 5375–5380, 2004. [PER 05] PERUSIN S., MONCEAU D., ANDRIEU E., “Investigations on the diffusion of oxygen in nickel at high temperature by SIMS analysis”, Journal of the Electrochemical Society, vol. 152, pp. E390–E397, 2005. [PRI 17] PRILLIEUX A., Hydrogen and water vapour effects on oxygen solubility and diffusivity in high temperature Fe-Ni alloys, PhD thesis, INP Toulouse, 2017. [SCO 99] SCOTT P.M., “An overview of internal oxidation as a possible explanation of intergranular SCC of alloy 600 in PWR”, Proceedings of the 9th international symposium on environmental degradation of materials in nuclear power systems-water reactors, 1999. [STE 09] STEINBACH I., “Phase-field models in materials science”, Modelling and Simulation in Materials Science and Engineering, vol. 17, no. 7, p. 073001, 2009. [TER 11] TER-OVANESSIAN B., Etude comparative de différents superalliages base Ni pour ressorts de système de maintien, PhD thesis, INP Toulouse, 2011. [TER 12] TER-OVANESSIAN B., POQUILLON D., CLOUE J.M. et al., “Influence of local mechanical loadings on the oxidation assisted crack initiation of alloy 718”, Materials Science and Engineering A, vol. 533, pp. 43–49, 2012. [VIL 14] VILLANI A., BUSSO E.P., AMMAR K. et al., “A fully coupled diffusionalmechanical formulation: Numerical implementation, analytical validation, and effects of plasticity on equilibrium”, Archive of Applied Mechanics, vol. 84, nos 9–11, pp. 1647–1664, 2014.
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[VIL 15] VILLANI A., A multi-physics modelling framework to describe the behaviour of nano-scale multilayer systems undergoing irradiation damage, PhD thesis, Mines ParisTech, 2015. [WEI 92] WEISS J., Endommagement en viscoplasticité cyclique sous chargement multiaxial à haute température d’un acier inoxydable austénitique, PhD thesis, Ecole des Mines de Paris, 1992. [WEI 93] WEISS J., PINEAU A., “Fatigue and creep-fatigue damage of austenitic stainless steels under multiaxial loading”, Metallurgical and Materials Transactions A, vol. 24, pp. 2247–2261, 1993. [WOO 06] WOODFORD D.A., “Gas phase embrittlement and time dependent cracking of nickel based superalloys”, Energy Materials, vol. 1, pp. 59–79, 2006.
15.1. Introduction Oxidation-assisted cracking (OAC) is a particular type of damaging process called “dynamic embrittlement”. Dynamic embrittlement is a fragile rupture process which implies the ingress, diffusion and segregation of embrittling elements at the grain boundaries leading to their debonding. These embrittling elements may be provided by the material itself, as in the case of sulfur in steels or tin in copper alloys, or by the material’s environment, notably oxygen’s partial pressure. One of the most-studied cases is OAC for nickel-based alloys for which the penetration and segregation of oxygen, often associated with intergranular oxidation, change the crack propagation mode from transgranular to intergranular and significantly increase the crack propagation rate [KRU 04, WOO 05, CHA 15]. In the aeronautical industry, as in the nuclear industry, this phenomenon is responsible for numerous cases of in-service cracking [HOC 94, CHA 97]. To study the propagation rate’s susceptibility to the chemical environment, numerous mechanical tests are conventionally performed in laboratory air or a vacuum. Figure 15.1 [PED 82] shows the propagation rates measured during low-frequency fatigue-creep tests (trapezoidal mechanical cycle: 10s–300s–10s) in the case of alloy 718 tested at 650°C in different environments. It is worth mentioning that oxidation has a significant effect on the crack propagation rate. In addition to the change in the propagation rate, the cracks propagate via the grain boundaries in air or the core of the grains in a vacuum. It should be noted that intergranular cracks might propagate at a maximum speed which may reach values of µm/s. The latter observation demonstrates the scientific Chapter written by Benoît TER-OVANESSIAN, Aurélien VILLANI, Éric ANDRIEU and Samuel FOREST.
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problems related to OAC. It is effectively necessary to take into account the oxidation modes, their kinetics, the local mechanics and the metallurgical state near the defect in order to propose physical modeling which is well underpinned.
Figure 15.1. Effect of oxidation and microstructure on the crack propagation rate in fatigue-creep test at 650°C for alloy 718
The time dependence for the OAC phenomenon is intrinsically linked to the temperature, the metallurgical state, the oxygen partial pressure, the oxide type and growth mode, the diffusivity of the weakening species within the grain boundary (increased or not by the local stress state) and finally the stress level (macroscopically and locally). Figure 15.2 illustrates an important point relative to the effect of oxygen’s partial pressure on the crack propagation rate at constant ΔK. It can be noted in Figure 15.2 that the propagation rate does not vary linearly with oxygen’s partial pressure and that a pressure threshold exists beyond which the propagation rate varies abruptly. This behavior is due to a change in the oxidation mode [GER 95].
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Figure 15.2. Evolution of the crack propagation rate at 650°C according to oxygen’s partial pressure at constant ΔK for different fatigue cycles
Mechanical loading plays multiple roles in the damage mechanism. It may act as a motor (tensile) or brake (compression) for the diffusion by acting on the flux and the solubility of the weakening species. In these interactions, the specific role of the mean stress or the deviatoric part of the stress field has to be considered. Mechanical loading may also act on the growth kinetics and the distribution of the growth stresses associated with oxide formation. Furthermore, when mechanical loading leads to local plastic or viscoplastic deformation, a segregation of impurities transported by the dislocations might lead to modification of the grain boundaries’ metallurgical state, and then to its sensitivity to intergranular oxidation. Reciprocally, in addition to the oxygen segregation at grain boundaries, their oxidation (the formation of intergranular oxide) plays an important role in the rupture. The geometry of the oxide tip allows us to test the grain boundaries toughness [AND 92]. The latter phenomenon may be considered as a geometrical assistance for the initiation and propagation of cracks. The relative contribution of these different factors to damage and their time dependence leads to various cases of OAC which are mostly differentiated by their cracking rates. When OAC occurs, a range of average propagation rates from a few mm/h to a few nm/h is covered [CHA 97]. Without being controlled systematically
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by a single mechanism, the different cases of OAC can be separated following a comparative analysis between the kinetics of fundamental mechanisms and the time scale associated with OAC rupture. Therefore, the objective of this chapter is not to focus our attention on one type of mechanism but rather to extend the subject by presenting different mechanisms, which are to various degrees potentially involved in OAC depending on the local conditions for mechanical and physicochemical loadings. 15.2. Multiprocess assistance: oxidation’s role Oxidation-assisted cracking’s complexity involving the contribution of chemical assistance, mechanical assistance and synergies makes it important to take into account the different mechanisms or sequences of mechanisms in order to understand the damage process, its kinetics and its mitigation. As an illustration, Figure 15.3 presents the evolution of the crack propagation rate during the superposition of a fatigue cycle and an oxidation cycle [MOL 97]. It is worth mentioning that the propagation rate only changes when the environmental cycle is superimposed to the ascending part of the mechanical cycle. This behavior demonstrates firstly the strong coupling between the local mechanical state and the damage induced by the oxide growth and secondly, the beneficial effect of unloading on the resistance to OAC.
Figure 15.3. Evolution of the crack propagation rate in low frequency fatigue according to the relative position of the environmental cycle with the mechanical loading cycle (alloy 718 at 650°C)
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Assuming that oxidation acts as the main chemical assistance, distinctions must be made to understand its consequences on the OAC phenomenon. Indeed, generally, oxidation of a metal at high or intermediate temperatures is classically described and studied according to three designations: external oxidation, internal oxidation and intergranular oxidation. The penetration at the grain boundary of “free” oxygen (oxygen in the solid solution) may be included in the latter process. Depending on the nature of the environment and the exposition temperature, these oxidation mechanisms are promoted to different extents and affect the material’s life via the mechanisms which are specific to them. 15.2.1. External oxidation As the external oxidation mechanisms have been detailed in a previous chapter, only some notions will be addressed here. The formation and the growth of an external oxide are due to an oxide-reduction reaction where the metal is oxidized and the oxygen is reduced. The formation of these oxides depends on the temperature, partial pressure in O2 and presence of water/steam or other aggressive gases. It is therefore mainly a thermodynamically controlled process. Two growth modes govern the thickening rate of the oxide: when the growth is controlled by the diffusion of cations, the oxide formation is located at the oxide-environment interface; when the growth is controlled by the diffusion of anions, the oxide formation is then located at the metal–oxide interface. These two modes may sometimes coexist. Nevertheless, in both cases, the metal–oxide interface mobility is a crucial issue. Indeed, the metal-oxide interface is displaced during oxidation in order to support the metal’s consumption (cationic growth) or to generate the space to accommodate the expansion in volume due to the oxide formation at the interface (anionic growth). This mobility must be included during the modeling of these phenomena. In these two situations, it is principally the flux of vacancies which will assure the mobility of the interface, provided the latter is not anchored by structural effects due to the geometry of the component or defect. For the case of cationic growth, when the interface is pinned, the injection of vacancies in the substrate leads to a supersaturation in vacancies. This supersaturation will lead to the formation of pores or cavities in the substrate, or even the recovery of the deformation structures by climbing dislocations in order to maintain the concentration of vacancies in the substrate at thermal equilibrium [PER 04a, PER 05]. Figure 15.4 illustrates the cavitation phenomenon induced by the injection of vacancies. For the case of anionic growth, the pinning of the external oxide layer restrains the setting of the flux of vacancies from the substrate towards the metal–oxide interface needed to generate the space for the oxide’s formation. In this case, vacancy sources activated within the metal are dislocations (climbing) and porosity when they preexist. These mechanisms, when they occur, create local
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relaxation of the stresses via the recovery of deformation structures. When they do not occur, the substrate is subjected to growth stresses, which may mechanically affect the geometry of the structural component. Nevertheless, these vacancies may contain oxygen and thereby not only lose their status as a thermal vacancy but also contribute to oxygen diffusion via substitutional sites. At the bottom of a grain boundary, these same mechanisms have specific consequences according to the grain boundary’s geometry and its disorientation. In the latter case, the contribution of external oxidation to OAC based on oxygen enrichment of the grain boundary will initiate intergranular oxidation.
Figure 15.4. Formation of an intergranular cavity in a strip of nickel oxidized on the two faces under laboratory air at 1000°C during 48h [PER 04b]
Furthermore, preferential oxidation of certain alloying elements has the consequence of generating a zone affected by oxidation in the substrate within which the chemical composition of the material is significantly modified. Thereby, the local metallurgical state is modified in terms of the precipitation state and the matrix lattice parameter. The consequences of preferential oxidation on the mechanical behavior of the affected zone may also slow down OAC by relaxing the local stresses or blunting the crack tip. 15.2.2. Internal oxidation In some cases, external oxidation is accompanied by internal oxidation. The preferential oxidation of certain elements in the matrix is indicative of penetration and diffusion of oxygen within the metallic material [SCO 99]. Beyond the “marker” effect by the precipitation of oxides, internal oxidation will also induce a
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change in the substrate’s chemical composition. Furthermore, like the concerns raised for external oxidation, the expansion in volume due to oxide formation may be accommodated by viscoplastic deformation or by condensation of vacancies coming from various sources.
Figure 15.5. Internal oxidation of an Fe-Ni (60%at )-Cr(7.5% at) alloy exposed for 100h at 1150°C in a RHINES pack [PRI 17]
Figure 15.5 [PRI 17] illustrates the latter phenomenon well. The internal oxide formation induces the formation of metallic protuberances at the sample’s surface. In this work, the oxygen’s partial pressure corresponding to the oxide’s dissociation does not permit the development of an external oxide layer. Consequently, the source of vacancies located at the sample’s surface and the flux of vacancies generates an opposite flux of iron and nickel atoms. The consequences of these phenomena on OAC are many. For example, the precipitation of oxides is hardening but the flux of vacancies associated with the formation of oxides are instead softening. It is therefore the local behavior of the zone affected by oxidation, resulting from the contribution of antagonistic phenomena, which will affect OAC.
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15.2.3. Intergranular oxidation Although the fundamental mechanisms involved are identical to those involved in so-called “external” oxidation (anionic or cationic growth, etc.), germination kinetics such as oxide growth at the grain boundaries are different from the behavior of the bulk material, as often observed with cross-sectional analysis. Indeed, local conditions are slightly different. In this case, the oxidation front grows at an interface, which may not have in its initial state the same chemical composition as the bulk material. Indeed, a grain boundary has a specific thermo-mechanical history resulting from the different metallurgical transformations, which monitor its chemical composition and its degree of disorientation. At the “substrate’s” scale, it no longer has the same crystallographic structure as the grain itself, which gives rise to diffusion coefficients that are significantly different with regards to the grain bulk. Grain boundaries generally react as short diffusion circuits. The consequences of intergranular oxidation are similar to external oxidation while being more damaging because they are at a more local scale. The critical level of the damage mainly depends on the grain boundary geometry and the depth of oxide penetration. The cohesion of this grain–grain interface can be affected in different ways: reduction of cohesion energy by oxygen segregation, edge effect due to oxide formation, selective pumping of an element involved in the formation of the external oxide (chromium depletion for example), formation of pores or cavities, etc. Even if these different items are present at the interface, intergranular crack will only initiate beyond a critical stress which, to be reached, requires a dedicated local mechanical loading. 15.3. From multiple processes to coupled processes Oxidation generates chemical, mechanical and microstructural modification of metallic materials at the core of the grain as well as at the grain boundary. The various consequences of this oxidation may act as brakes or motors to OAC crack initiation and propagation. However, as mentioned previously, grain boundary rupture can occur only if a critical stress is reached locally. The coupling between the chemical process and mechanical loading is therefore necessary for OAC cracking. The study of these synergies is then required in order to understand OAC. 15.3.1. Mechanical-diffusion coupling In the previous paragraphs, different mechanisms influencing OAC have been detailed. Not only various and numerous, these processes are often also interdependent. The importance of species diffusion has been highlighted in the description of the mechanisms. However, the mechanical state at the crack tip or
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generated by a macroscopically mechanical loading may interfere with it. To better understand the mechanics-diffusion coupling, its modeling is therefore essential. A well-known example is Cottrell’s atmosphere [COT 75]: under its own stress field, a dislocation locally reorganizes the distribution of vacancies (Figure 15.6) and solutes of a material. Although known for a long time, this phenomenon is still the object of studies, whether at the atomic scale [MIS 16], or at the continuum scale [CAH 13].
Figure 15.6. Vacancy concentration field around an edge dislocation in an infinite medium, given by a continuum theory, and normalized by the homogeneous concentration at thermal equilibrium. Since the analytical solution of the stress field is singular, the dislocation core has been omitted. For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
In this paragraph, the numerical approach considers the continuum scale. Indeed, the conventional thermodynamic framework for continuum mechanics is perfectly adapted to the simulation of mechanical-diffusion coupling. The key idea here is to introduce into the strain components an eigenstrain depending on the local concentration fields. For example, in the case of only one chemical species, the eigenstrain is expressed as: 𝜀∗ 𝑐 = 𝜂 𝑐 − 𝑐
[15.1]
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where c is the local concentration of the species considered, 𝑐 is its concentration at thermal equilibrium and η the coupling factor. In general, η is a tensor proportional to the identity, or diagonal. As for a thermo-mechanical coupling, a local variation of concentration leads to a local volume change. Indeed, under- or over-saturation of alloying elements, or of vacancies, locally disturbs the crystal lattice. This coupling term having been introduced, conventional application of thermodynamics for continuum mechanics leads directly to an expression of the flux vector of the type: 𝐽 = −𝐷𝛻𝑐 + 𝑓 𝜂 𝛻 𝑡𝑟𝑎𝑐𝑒 𝜎
[15.2]
which is nothing less than a modified Fick’s law, with D the diffusion coefficient. Note that the second term corresponds to the contribution of the mechanics in the diffusion potential, and is proportional to the coupling term η, as well as to the gradient of the trace of the stress tensor. It is noted that the general theory also gives the concentration field at equilibrium, according to the boundary conditions, but the reader is referred to [VIL 14] for more information. The influence of mechanics on the diffusion is therefore obvious. In principle, the theory mentioned previously also gives the influence of the concentration field on the stress field at equilibrium, via the use of elastic constants derived at constant chemical potential. In practice, with typically low concentrations, this effect is negligible. To use the example of the dislocation, the ratio of the hydrostatic pressure fields given by a coupled and uncoupled elastic theory is typically 0.999. Except for particular cases, irradiation for example, it is possible to use conventional elastic constants in a coupled problem. The equations for the model briefly introduced above can be solved relatively simply by finite elements method [VIL 14]. In the following example, a hollow and notched cylindrical test specimen is filled with hydrogen at constant partial pressure, Figure 15.7(a). In the absence of mechanical loading, hydrogen diffuses until the concentration is homogeneous in the overall specimen. However, if the test specimen is loaded with tensile or compressive stress along the rotating axis, the notch will play the role of a stress concentration. Following equation [15.2], the concentration at equilibrium will be therefore heterogeneous. For tensile tests, it is in the bottom of the notch where the concentration is the highest, with a value about 1.05 times higher than elsewhere within the test specimen for this example (Figure 15.7(b)). In the case of a crack, the mechanical solution in elasticity is singular, just like for the dislocation given as an example at the start of this section. Even if the use of
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a viscoplastic model removes the singularity, the crack tip’s mechanical field remains strongly nonlinear, as we will see in section 15.3.3. Therefore, the local concentration at the crack tip can be strongly modified by the crack’s stress field, and thus significantly impacts OAC kinetics.
Figure 15.7. An interstitial concentration of hydrogen is imposed on the internal face of a notched hollow cylindrical specimen, modeled in a). The specimen is loaded mechanically along its rotating axis, and the concentration at equilibrium in the section of the specimen is presented in b) for tensile loading, and in c) for compressive loading. For a color version of this figure, see www.iste.co.uk/blanc/ coupling.zip
15.3.2. Coupling intergranular mechanical-diffusion-oxidation behavior As previously shown, the diffusion of the solutes is modified when a stress field is applied, highlighting the mechanical-diffusion coupling which may be involved in OAC. Observed macroscopically, this contribution certainly locally influences the grain boundary (where the diffusivity of the species is already different) or the crack tip. However, the chemical species’ mobility in a material under stress can also be modified by plastic deformation, or plastic flow. Solute-dislocation interactions, whether they are viscous drag type or pinned/released type, are known to be responsible for irregularities in mechanical behavior such as dynamic strain ageing, Luders or Portevin-Le Chatelier (PLC) effects. Thermally activated, these different deformation modes also depend on the strain rate and metallurgical parameters,
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which jointly influence the solutes’ diffusivity and the mobility and localization of dislocations. When these deformation modes are observed, the associated interactions play a role in the OAC process. Numerous works have been performed in the last few years to study the interaction between deformation modes and OAC sensitivity for nickel-based superalloys [FOU 00, FOU 01, GAR 08, DEL 07]. Their first observation is that during tensile tests in air for temperatures between 300°C and 700°C, the fracture surface of the alloy 718 presents no fragile intergranular zone when plastic instabilities are observed on the tensile curves. Conversely, the absence of these instabilities is directly linked to the emergence of fragile intergranular zones on the fracture surfaces, in the same temperature domain. These tests have also been performed in an inert atmosphere, and do not lead to fragile rupture. There is therefore a robust correlation between the OAC’s occurrence and the local plastic deformation modes. Figure 15.8 shows a temperature–strain rate map for alloy 718. The intergranular rupture–transgranular rupture frontier coincides with the dynamic strain ageing–PLC frontier. Different mechanical or chemical scenarios can be proposed to explain this damage mode. The PLC flow instabilities may be considered differently: propagation of a deformation band, limited germination and propagation of several bands and random germination of small bands. The amplitude of the instabilities on the tensile curve allows us in general to determine the type of the PLC. In general, this behavior is a result of the strain’s localization at the scale of the test specimen, which leads to a reduction of the material’s elongation in the case of rupture in an inert atmosphere. The deformation in a strip is very rapid and rather homogeneous. Consequently, the local strain rate is not very compatible with oxidation kinetics. Furthermore, a decrease in the stress is observed using a hard tensile testing machine (imposed displacement), making it less likely that defects will be opened which were not open before. This type of scenario applied in the plastic zone of a propagating crack tip leads to very rapid relaxation of the stress field at the tip of the defect and therefore inhibits OAC. It is also possible to imagine that this flow instability effect applied to a defect could inhibit crack initiation. However, whilst the loading mode is associated with dynamic strain ageing, the rupture becomes intergranular in an oxidizing atmosphere but remains transgranular in an inert atmosphere [MAX 16]. In this case, plastic strain is supposed to be distributed on the entire gage length of the specimen at the macroscopic scale. However, at the polycrystal scale, we believe that deformation heterogeneities generate significant intergranular stresses on certain grain–grain interfaces. Additional studies have shown that for these modes of plastic flow, it was necessary to locally associate a critical cumulated strain allowing the cohesive rupture of the grain boundary (Figure 15.8) [TER 12].
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Figure 15.8. Experimental mapping of the deformation modes and the rupture modes for alloy 718 in the temperature field 550–700°C. The cumulated plastic strain necessary for intergranular cracking in air is reported (see [TER 11]). For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
A more chemical approach establishes the relationship between the transport of solutes to the grain boundaries by dislocations, deformation modes and intergranular rupture. In the dynamic strain ageing regime, without any occurrence of PLC effects, the viscous drag of the solutes towards the grain boundaries and the zones with a high stress concentration can locally modify the chemistry of the grain boundary, already affected by the penetration of oxygen or oxide, and induce decohesion of the latter. In the case of a PLC effect, the pinning/unpinning mechanism does not lead to such grain boundary enrichment and local interactions with the dissolved oxygen. This chemical approach does not contradict the mechanical approach. The notion of damage accumulation can also be seen as an accumulation of solutes at the grain boundaries. The chemical species suspected to weaken the grain boundaries are the interstitial elements C, H, N, which react easily with oxygen. However, a recent study dedicated to this alloy, using internal friction measurements, showed that molybdenum was the alloying element most likely to induce plastic instabilities in the temperature range explored [MAX 16, MAX 18]. This work
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clearly highlights the coupling between mechanical behavior, chemical species’ mobility, intergranular oxidation and OAC sensitivity. The assumptions made in order to explain this type of damage warrant confirmation via the acquisition of new experimental or numerical data. 15.3.3. Simulation methods for mechanical-oxidation intergranular coupling The fascinating microstructure in Figure 15.9 contains all the features necessary for modeling oxidation and plasticity-assisted cracking. We see that the oxidation takes some preferred routes which will be possible locations for crack initiation: the intense plastic slip bands and the grain boundaries. The V shape of the oxidation tip at the slip bands shows the importance of oxygen’s accelerated diffusion in these zones and the angle of the tip providing information on the associated interface energy. Diffusion, oxidation, elasticity, viscoplasticity and damage are the mechanisms to couple in a unified modeling approach. In a global approach, multiple chemical reactions, which are linked to complex oxides and the electrochemical aspects, should also be added.
Figure 15.9. SEM microstructure for a 316L steel loaded in torsion during a fatigue-creep test and then exposed for one hour for oxidation, according to [WEI 92]. See also [WEI 93]
The phase field method is a thermodynamic framework for modeling the evolution of the phase change fronts and their interaction with diffusion and mechanics [STE 09, FIN 10]. To illustrate the potential as well as the complexity of this approach, we detail the example of chromium oxide formation in a
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(quansi-)stainless steel 316L. The model’s chemical part is described by the density of Helmholtz free energy which depends on the species concentrations (to simplify, O, Cr and Fe) and by the dissipation potential associated with the diffusion of these elements. The phase field is a variable passing smoothly from 0 in steel to 1 in the oxide. Its evolution is governed by the Allen–Cahn equation (also called Ginzburg–Landau), which includes a diffusive term and chemical driving forces for the front’s progress, as well as a mobility parameter for the interface. The internal oxidation front is treated as a diffuse interface whose thickness is regulated by the associated volumetric free energy and an equivalency with the interface energy. It is a characteristic of the phase field method in relation to the treatment of perfect interfaces for which changes in the shape, intersections and connections are difficult to handle. Intense slip bands and grain boundaries are diffusion short-circuits for oxygen and remain so once preferential oxidation paths have been established. This fact is taken into account in the simulation by the use of amplified diffusion coefficients in these zones. Figure 15.10 illustrates the formation of an intergranular oxide tip between two grains as well as the resulting depletion in chromium in the underlayer matrix. These concentration fields can be directly compared to experimental fields measured in SEM or TEM for example. The shape of the tip is not imposed; it results from the competition between interface energy, diffusion and also mechanics due to stresses generated by the eigenstrain associated with the oxide growth. This last parameter is essential for coupling the chemical thermodynamics with the mechanics but it is still badly known. We cannot be satisfied with the Pilling–Bedworth relationship, which takes into account the change in volume during the passage from the metallic phase to the oxide phase, because it leads to the calculation of stresses which are much higher than the residual stresses measured in the oxides [HUN 02]. The complexity of the process for oxide formation which results from the accommodation of the coherence stresses instead leads to an anisotropic eigenstrain, as proposed by Parise et al. [PAR 98], including components identified by an inverse approach starting from the stress measurements. The identification of the anisotropic eigenstrains via phenomenological or physical procedures is a major issue for model development in this field. The stresses generated around the oxidation tip are sufficiently high to trigger the activation of slip systems in the grains concerned. The theory and the numerical practice of crystal viscoplasticity are applicable in this case, as shown in Figure 15.11 (shown on the right), which gives the accumulated plastic slip field in the growth zone by assuming an ideal orientation of the FCC grain. Figure 15.11 (shown on the left) shows furthermore that hydrostatic stress is rather high which prompts us to utilize damage or local rupture criteria to predict crack initiation and propagation. Because of the tip’s complex geometry, we observe a tensile zone in front of the oxidation tip as well as at the back, even in the absence of external
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loading. The model then assumes a cracking process at the front contributing to intergranular damage, as well as at the back promoting the accelerated penetration of oxygen, or other aggressive species. The simulation of generalized intergranular damage is part of the toolbox of numerical simulations for crystalline plasticity coupled with diffusion, according to the work of Diard et al. [DIA 02] and Musienko and Cailletaud [MUS 09].
Figure 15.10. Chromium depletion during the oxidation of a grain boundary in a 316L steel [DER 15]. For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
Figure 15.11. Hydrostatic pressure field in the oxide and the substrate (to the left, the white corresponds to tensile values); cumulated plastic strain field in the substrate, the red exceeding one percent of plastic strain [DER 15]. For a color version of this figure, see www.iste.co.uk/blanc/coupling.zip
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15.4. Conclusion Although these couplings or synergies are observed experimentally and analyzed at increasingly fine scales, their modeling by numerical methods, which may rely on experimental validation, are an indispensable contribution to the understanding and discrimination of OAC mechanisms. This contribution provides us with the opportunity to firstly simulate and correctly predict the lifespan of structural elements and secondly, to find new ways of developing and/or designing in order to secure the structures exposed to this type of damage. The approach based on the phase field method, allowing us to couple the progress of an oxidation front with the diffusion of chemical elements, crystalline plasticity and damage, is extremely promising, even if the results remain qualitative nowadays. Considerable work is necessary to gain realism from the viewpoint of the coupling mechanisms involving microstructure, mechanical state and oxidation. 15.5. References [AND 92] ANDRIEU E., MOLINS R., GHONEM H. et al., “Intergranular crack tip oxidation mechanism in a nickel-based superalloy”, Material Science and Engineering A, vol. 154, pp. 21–28, 1992. [CAH 13] CAHN J., “Thermodynamic aspects of Cottrell atmospheres”, Philosophical Magazine, vol. 93, pp. 3741–3746, 2013. [CHA 97] CHASSAIGNE J.C., Fissuration à haute température du superalliage base nickel N18 élaboré par métallurgie des poudres, Etude du couplage mécanique environnement en point de fissure, PhD thesis, Ecole des Mines de Paris, 1997. [CHA 15] CHAN K.S., “A grain boundary fracture model for predicting dynamic embrittlement and oxidation-induced cracking in superalloys”, Metallurgical and Materials Transactions A, vol. 46, pp. 2491–2505, 2015. [COT 75] COTTRELL A., An Introduction to Metallurgy, 2nd ed., Edward Arnold, London, 1975. [DEL 07] DELEUME J., Facteurs métallurgiques et mécaniques contrôlant l’amorçage de défauts de corrosion sous contrainte dans l’alliage 718 en milieu primaire des réacteurs à eau sous pression, PhD thesis, INP Toulouse, 2007. [DER 15] DE RANCOURT V., Modelling the oxidation of polycrystalline austenitic stainless steels using a phase field approach coupled with mechanics, PhD thesis, Mines ParisTech, 2015.
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