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A mechanism governing oxidation-assisted low-cycle fatigue of superalloys
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Acta Materialia 57 (2009) 2969–2983 www.elsevier.com/locate/actamat
A mechanism governing oxidation-assisted low-cycle fatigue of superalloys A.G. Evans a, M.Y. He a,*, A. Suzuki b, M. Gigliotti b, B. Hazel c, T.M. Pollock d a
Materials Department, University of California at Santa Barbara, Santa Barbara, CA 93106, USA b General Electric Global Research, Niskayuna, NY 12309, USA c General Electric Aviation, Cincinnati, OH 45215, USA d Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109, USA Received 9 December 2008; received in revised form 28 February 2009; accepted 28 February 2009 Available online 1 April 2009
Abstract A model capable of characterizing oxidation-assisted low-cycle fatigue is described. It involves the following steps. After a few strain cycles, because of creep, a tensile stress develops during the de-straining phase of the cycle. This stress opens cracks present in the material and exposes the surfaces to the atmosphere, causing thermally grown oxide (TGO) growth. Dilatation takes place upon converting the alloy to oxide, with an associated strain rate that induces a compressive growth stress. Thereafter, during the re-straining phase of the cycle, transverse extension of the substrate induces in-plane tension in the TGO, which ‘‘pushes” the TGO into the substrate along the crack front. Finite element simulations of this process have been presented that predict crack growth per cycle, da/dN, comparable with experimental measurements. Trends in da/dN with the TGO dilatation rate and the creep strength of the superalloy have been elucidated. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Low-cycle fatigue; Superalloys; Thermally grown oxide; Finite element
1. Introduction Airfoils used in gas turbines incorporate coatings for oxidation and thermal protection [1–4]. The coatings can affect the integrity of the airfoil since thermomechanical fatigue initiates at the component surface. Over the spectrum of temperatures, strain states and stresses generated, one of the most detrimental fatigue mechanisms is that upon isothermal, sustained peak low-cycle fatigue (splcf) [5–7]. This mechanism occurs at temperatures sufficient to induce creep of the superalloy, upon exposure to cyclic compressive displacements under oxidizing conditions (Fig. 1a). The creep at the peak strain partially relaxes the stress, whereupon tension develops as the displacement returns to zero. Indeed, after multiple cycles, the tensile stress and compressive stresses attain similar levels *
Corresponding author. Tel.: +1 805 893 7166; fax: +1 805 893 8486. E-mail address: [email protected] (M.Y. He).
(Fig. 1b). Actual measurements are presented elsewhere [8]. The development of tension, in conjunction with oxidation, is regarded as a primary motivation for fatigue, in accordance with the following generalities. Multiple crack-like imperfections develop from the surface of the bond coat [9–12], then propagate on a cycle-by-cycle basis, and evolve with cycling in three distinct stages, illustrated in Fig. 2. In stage I, they extend along the surface, are Vshaped in profile, and tend to arrest at the interdiffusion zone (IDZ) between the coating and the substrate. After further cycling, during stage II, the cracks enter the IDZ, and accelerate with reduced included angle. Once they reach the substrate, in stage III, some penetrate, and accelerate again, in accordance with a constant-width configuration. There is also a subsequent, stage IV, when a mixed-mode crack forms and extends along the {1 1 1} planes [12–15]. Once this happens, most of the cyclic life has been consumed, and consequently this stage is not emphasized in this paper. Throughout, a thermally grown
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.02.047
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Nomenclature a E hTGO n N S tstrain d epl e_
crack half length Young’s modulus TGO thickness creep exponent strain-hardening exponent crack spacing time period for imposing the strain during each cycle crack opening on surface plastic strain strain rate
oxide (TGO) layer forms on the crack surfaces, and exerts a central influence on the fatigue mechanism. In stage I, the TGO is largely a-Al2O3. Thereafter, refractory metal and ternary oxides form in addition to alumina, causing the oxidation kinetics to accelerate as the crack progresses downward from the surface. The intent of this paper is to examine the mechanics of cracks subject to compressive strain cycling and thereby ascertain the roles of the oxide layer as well as the constituent creep strengths on the crack extension per cycle. The paper is organized as follows. A synopsis of the salient measurements and observations of fatigue cracking is presented. Thereafter, a connection between the fatigue mechanism and TGO rumpling [16–20] is postulated, along with a synopsis of the associated phenomena. The fatigue model is presented and numerical results generated for stage III to demonstrate that the measured phenomena can be predicted without any additional mechanics and physics. The features that emerge are used to elucidate
substrate creep rate e_ sub reference strain rate for creep e_ 0 egrowth unconstrained elongation strain for TGO during growth # included angle of V-shaped crack in stages I and II rgrowth stress in TGO as it thickens reference stress for creep r0 von Mises stress req yield strength of substrate alloy rsub Y rbc yield strength for bond coat Y
the parameters that affect the crack extension per cycle. Numerical results for stage I are provided that relate fatigue to the properties of the bond coat. 2. Synopsis of measurements and observations Cylindrical test specimens of the superalloy Rene N5 (12 mm diameter) with a b-phase, aluminide bond coat were subjected to isothermal testing at 1090 °C. Axial (xdirection) compressive displacements were imposed, corresponding to a strain amplitude of 0.35%. Tests were interrupted after various fractions of the cyclic life and characterized. The evolution of the cyclic stress–strain response with cycling is presented schematically in Fig. 1. A detailed description of the stress evolutions and the crack growth process will be published elsewhere [8]. Note that, at these temperatures, the Young’s modulus is E 74 GPa (loading along [0 0 1]) and that the stress relaxation during each hold period is small relative to the total stress.
Fig. 1. Schematics of the applied straining conditions associated with sustained peak low-cycle fatigue. (a) The compressive strains imposed and trends in the stresses induced as cycling proceeds. (b) The stress–strain loops and their evolution with cycling.
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Fig. 2. Schematics of the fatigue cracks during their three stages of development. (a) V-shaped stage I cracks contained within the bond coat. (b) Stage III parallel-sided cracks within the substrate.
Multiple cracks are apparent along the surface after less than 10% of the cyclic life (Fig. 3). They have small opening displacement, d 2 lm (or less), length 2a 100 lm (or less), spaced apart by s 100 lm. After about 50% of life, the openings are much larger, d 20 lm, and several contiguous cracks have coalesced. Cross-sections (Fig. 4) reveal that, with time at temperature, the bond coat converts to two-phase b/c0 , while the IDZ becomes largely c0 . Moreover, under these test conditions, stage I persists during the first 50% of life, with the cracks invariably in the b phase with included angles of 20 6 # 6 30 (Fig. 4a). The TGO is relatively uniform, thickness hTGO 3 lm
(Fig. 5). Thereafter, up to about 70% of life, in stage II, the cracks penetrate the c0 -phase in the IDZ, and have an included angle # 15° (Fig. 4b). In stage III, the cracks propagate in the substrate with parallel sides and TGO thickness again, hTGO 3 lm (Fig. 4c). The crack growth rate, da/dN, increases dramatically at the onset of stage III (Fig. 6). 3. Connections to TGO rumpling Prior assessments of TGO rumpling upon thermal cycling have revealed crack-like imperfections in the bond
Fig. 3. A scanning electron micrograph of the top of the bond coat after about 10% of cyclic life at 1090 °C, showing stage I cracks extending along the surface normal to the strain axis.
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Fig. 4. Scanning electron images of cross-sections normal to the surface showing the fatigue cracks in profile. (a) Stage I crack in the bond coat. (b) Stage II crack that has penetrated the IDZ. (c) Stage III crack within the substrate.
coat (Fig. 7) that have features similar to stage I cracks. These features have been found experimentally [19] and replicated by numerical simulations upon finite-element
implementation of a rumpling code [18]. Because of this similarity, it is hypothesized that splcf is another manifestation of the same mechanism, albeit subject to quite dif-
Fig. 5. A higher-resolution SEM image of a stage I crack highlighting the TGO layer on its surfaces.
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Fig. 6. (a) The depth of the largest fatigue cracks as a function of the fraction of cyclic life. (b) The growth rate of the same crack, estimated by taking the derivative of a best fit through the datum points on (a).
ferent loading conditions. To explore this possibility, some of the basic phenomena involved in rumpling are summarized. The following elements are especially important. (i) During rumpling, the TGO not only thickens but also elongates. (ii) As the undulations grow, the downward displacements are accommodated by creep deformation of the bond coat.
3.1. TGO thickening and lateral straining Oxidation of the bond coat occurs by the formation of polycrystalline, columnar a-Al2O3 [2]. While most of the new oxide forms at the interface with the bond coat by inward diffusion of O, an outward counter-flux of Al causes some new a-Al2O3 to form along the through-thickness
Fig. 7. Cross-section of a crack-like perturbation caused by TGO rumpling during thermal cycling. (a) A cross-sectional image (courtesy Tolpygo and Clarke [11]). (b) Simulation conducted using a FE version of the rumpling code [9].
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Fig. 8. (a) Fractured cross-section of an alumina TGO illustrating the inner, columnar portion of the oxide formed by inward diffusion of O and the outer, equiaxed portion formed by outward diffusion of Al [2]. (b) Illustration of an alumina TGO on reoxidation after smoothly polishing the TGO formed in the first oxidation step. New oxide forms along the grain boundaries of the initially formed TGO and the amount increases with further oxidation [2].
grain boundaries, as well as at ridges along the surface [21,22] (Fig. 8). That formed on the grain boundaries must be accommodated by in-plane straining of the neighboring grains, at strain rate e_ growth [16–18,23] (Fig. 9). Hence, as shown on the left of Fig. 9, when a counter-flux of Al and O meets the internal grain boundaries with the new a-Al2O3 the neighboring grains must be compressed to accommodate the volume. Thus, e_ growth is a measure of the rate at which the TGO would elongate if theoretically detached from the bond coat and substrate. The a-Al2O3 responds to this strain rate by causing a compressive
growth stress, rgrowth (Fig. 9) [24,25]. The magnitude of this stress is governed by the creep mechanisms operating in aAl2O3 at the test temperature. This has been ascertained by measuring rgrowth in situ for the TGO as it grows on several different bond coats. This stress is of the order rgrowth 300 MPa [25] (Fig. 9, right), consistent with the deformation mechanisms for a-Al2O3 and with stress relaxation rates measured in a typical TGO [26]. When the surface is non-planar, the stress rgrowth exerts a downward pressure on the bond coat, causing it to creep [16,18], and resulting in the development of crack-like features [18].
Fig. 9. (a) When a counter-flux of Al and O meets on the internal grain boundaries to from new a-Al2O3 the neighboring grains must be compressed to accommodate the volume: e_ growth is a measure of the rate at which the TGO would elongate if theoretically detached from the bond coat and substrate. The a-Al2O3 responds to this strain rate by causing a compressive growth stress, governed by the creep mechanisms operating in a-Al2O3 at the test temperature. (b) An illustration of the stress in the TGO measured in situ in the synchrotron during a single thermal cycle from ambient to 1125 °C and back to ambient [2].
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Oxides also form on the IDZ and substrate. Their compositions differ, as already noted. The corresponding oxidation mechanisms and growth stresses have not been studied. However, these other oxides thicken more rapidly than a-Al2O3 [27], suggesting a larger elongation strain rate, e_ growth . However, they also creep more rapidly [28] (larger e_ creep ), so the resultant growth stress is indeterminate. It is treated as a parameter in the following model.
Table 2 Creep strengths of the substrate and coating layers.
3.2. Constituent creep rates
as the tension becomes substantial upon de-straining (after 10% of cyclic life, Fig. 1), the crack opens and the TGO elongates. To enable extension of the TGO-filled crack, outward creep flow must occur in the substrate around its frontal zone, similar to the flow during TGO rumpling [16,17,23]. The nature of this flow is affirmed in the following analysis [16,17]. Recall that the crack opening upon destraining is a direct consequence of the bulk creep deformation during the compressive hold, which relaxes the compressive stress. Indeed, direct evidence of uniform compressive creep throughout the sample is provided by the evolution of the precipitate morphology, with plates aligned parallel to the compression axis (Fig. 10). The principal features can be elucidated by using a plasticity model for the substrate and bond coat (with powerlaw hardening) and by invoking the reference stress method to convert to power-law creep [32]. In this method, the materials are assigned a yield strength and strain-hardening coefficient. The plastic deformations are elucidated in accordance with J2 plasticity theory, through the evolution of the von Mises stresses. The correspondence between the deformations that develop from plasticity and those induced by creep has been demonstrated for rumpling [16–18]. The viability is demonstrated below by comparing representative results based on plasticity and creep. Thereafter, for computational efficiency, trends are presented using the plasticity version. The predictions are obtained using a finite-element (FE) method: an analytical version could be pursued later, once the numerical results have delineated the phenomena. The FE code ABAQUS standard is implemented in-plane strain, with the meshes depicted in Fig. 11. The parameters are those presented in Table 1. For the plasticity analogue, the yield strength for the substrate and the strain-hardening exponents are those given in Table 1. In the simulations the crack advances through the elongation of the TGO elements in its wake, with small-scale plastic blunting occurring in the elements at the tip. Hence, the self-similarity of the crack growth in the steady state. It has not been necessary to use a crack tip rupture criterion.
Creep rates have been measured in both the substrate [29] and the bond coat [30]. In the steady state, they can be adequately represented by power-law behavior of the form: e_ s e_ 0 ðreq =r0 Þn , where req is the local von Mises stress, r0 is a reference stress, e_ 0 a reference strain rate, and n the creep exponent. Values for these parameters representative of the substrate at 1090 °C are summarized in Table 1. The corresponding creep rates for the bond coat are known [30] but are not used in this paper. The creep properties of alloys with compositions similar to those in the IDZ have been measured. They are intermediate between those of the b-NiAl bond coat and the superalloy substrate [29–31] (Table 2). 4. Substrate fatigue (stage III) The fatigue model proceeds from the rumpling model without any additional physics or mechanics, in accordance with the same numerical implementation. That is, to capture the in-plane (y-direction) straining in the TGO, in the model, e_ growth is imposed during those stages of the strain cycle when the crack is open. It is imposed at a uniform rate governed by the TGO thickness [16,17]. This strain rate causes creep of the TGO, replicated by ensuring that the von Mises stress never exceeds the growth stress, req 6 rgrowth . The ensuing response of the TGO is governed by the creep deformation of the surrounding material (bond coat, IDZ or substrate). Hence, creep deformations occurring around the tip accommodate elongation of the TGO, leading to a crack extension per cycle. The five parameters affecting the fatigue crack growth rate are thus: the TGO growth stress, rgrowth; the elongation strain rate experienced by the TGO as it grows, e_ growth ; the TGO thickness, hTGO; the creep characteristics of the surrounding material, n e_ sub e_ 0 ðr=r0 Þ ; and the crack length, a. The phenomenon is most readily visualized for stage III crack extension in the substrate (but is equally applicable to V-cracks in the bond coat and IDZ, described in Section 5). Specifically,
Composition
Creep strength (in MPa) at 1090 °C for creep rate 108 s1
Ref.
Rene´ N5 (substrate) Ni–6.3Al–15.1Pt–2Cr– 2.9Re–5.6Ta (wt.%) (IDZ) B2 (Pt,Ni)Al (bond coat)
110 60
[21] [23]
20
[22]
Table 1 Constituent material properties at 1090 °C.
TGO Substrate Bond coat
Yield strength, rY (MPa)
Young’s modulus, E (GPa)
Strain-hardening exponent, N
Strain rate coefficient, e_ 0 ðs1 Þ
Strain rate exponent, n
Creep reference stress, r0 (MPa)
300 50–200 10–40
400 74 74
– 10 10
– 4 109 –
– 10 –
– 100 –
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Fig. 10. Rene´ N5 sample with a crack extending through the IDZ into the superalloy substrate following fatigue loading with a compressive hold in the direction indicated by the arrows. Directional coarsening of the c0 precipitates with alignment parallel to the compression axis occurs due to bulk creep deformation in the superalloy during the compressive hold (Courtesy C. Brundidge).
4.1. The plasticity version The FE mesh used for the stage III calculations is depicted in Fig. 11, which shows the results after 20 cycles.
The red region is the TGO and the blue region the substrate. The horizontal region with a fine mesh represents the original location of the crack. Evidently the crack has extended to its new location after cycling. Within the
Fig. 11. (a) The FE mesh used for the stage III calculations after 20 cycles. The red region is the TGO and the blue region the substrate. The horizontal region with fine mesh represents the original location of the crack. Evidently the crack has extended to its new location upon cycling. (b) The near-tip region when the configuration is at zero strain and the crack is open. (c) The configuration when the applied strain is a maximum (crack closed except near the tip). The stage I simulations use a different mesh (not shown for brevity). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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context of the model (Fig. 11), a transverse (x-direction) cyclic displacement is imposed, at the boundaries, consistent with a compressive strain cycling between 0% and 0.4% (Fig. 1). The surface is free to displace in the y-direction. The zero displacement (dY = 0) plane is located in the substrate 5 mm from the surface. The TGO elongation strain is imposed at a constant rate, while the crack is open. This is achieved within ABAQUS by imposing an in-plane stress-free strain rate (similar conceptually to the way thermal expansion and phase changes are addressed). In the plasticity model, since the deformation is time invariant, assigning a strain/cycle (rather than a strain rate) suffices. Actual strain rates must be used in the creep model. The elongation strain range, 0 6 egrowth 6 0:1%=cycle, has been chosen because of its relevance to the prediction of rumpling rates [16–18]. In the simulations, this strain is imposed at a constant rate throughout the cycle. The growth stress is retained at, rgrowth 300 MPa [25], and the TGO thickness fixed at the measured value, hTGO 3 lm. The yield strength of the substrate is, rsub Y ¼ 100 MPa, and strain-hardening exponent, N = 10 (Table 1). All calculations are for 20 cycles with initial crack length a0 = 100 lm, such that the extension, Da, is small relative to a0. Calculations conducted for other a0 are expected to predict somewhat different da/dN: the associated trends are subject to a separate assessment. Unless stated otherwise, the results shown below are for a TGO strain, egrowth = 0.05%/cycle. The displacements presented in Fig. 11 indicate that the crack is open over a large portion of the strain cycle and, after a few cycles, remains open at the tip, throughout the entire cycle (Fig. 11c), though it closes over most of its length when fully compressed. The stresses induced at the boundaries over 20 cycles are presented in Fig. 12a. After a few cycles, steady state is reached with about equal tension and compression, with a hysteresis loop having a width, Dehyst = 0.05. The cycle-by-cycle extension of the
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crack front (Fig. 12b) reveals that, after a few cycles, the extension per cycle is invariant and (for this case), da/ dN = 26 nm/cycle, quite similar to that ascertained experimentally in stage III (da/dN = 28 nm/cycle for the corresponding crack length and similar test conditions) (Fig. 6). The profile of the crack front remains essentially unchanged as the crack opens and closes (Fig. 11). The corresponding behavior in the absence of an elongation strain rate, e_ growth ¼ 0 (Fig. 12b) confirms that continuous fatigue crack growth is linked directly to TGO formation. Additional results for other e_ growth indicate its specific influence on the crack growth rate (Fig. 13). This trend is to be interpreted from the perspective that e_ growth is composition dependent and thus affected by the distance from the surface. There is an associated influence of the substrate yield strength, with da/dN decreasing as rY increases (Fig. 13).
Fig. 13. Influence of the lateral strain rate of the TGO and of the substrate yield strength on the stage III crack extension per cycle.
Fig. 12. (a) The hysteresis loops calculated using the plasticity model. (b) Calculations of the change in crack length in stage III for a representative TGO elongation rate, e_ growth ¼ 0:1%=cycle, as well as for e_ growth ¼ 0, contrasting the two situations.
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Fig. 14. The stresses induced in the TGO during a strain cycle. (a) The in-plane stresses with and without a growth strain. (b) The concentrated rxx stresses ahead of the crack when closed and open.
Diagnosis of the stresses, strains and displacement during one cycle help to explain the phenomena governing crack extension (Fig. 14). The assessment starts with the crack closed, when the imposed compressive strain is at its maximum. Upon reducing the applied strain, the Poisson effect requires that the substrate retract along the crack plane, causing the TGO to develop in-plane compression (Fig. 14). Addition of the growth strain introduces further in-plane (y-direction) compression, causing the stress in the TGO to attain its maximum (300 MPa). Close to the front, the transverse compression in the TGO is larger, rtgo xx 600 Mpa (Fig. 14) because of the triaxiality. The corresponding stress in the substrate is also compressive, but smaller, rsub xx 120 MPa (Fig. 14). The plastic strain field has features representative of those around cracks, with the largest strains occurring within contours inclined at 30–40° to the crack plane (Fig. 15). In such [0 0 1] facecentered cubic single crystals, this inclination coincides with the orientation of the octahedral slip planes to the crack front, indicative of localized slip along these planes during fatigue [13,33]. These planes are ultimately the direction along which the cracks propagate in stage IV, when the mixed-mode stress intensity becomes sufficiently large [13]. Upon re-straining, the substrate lengthens along the crack plane, causing the TGO to develop in-plane tension (Fig. 14). The growth strain counteracts this tension such that the net in-plane stress in the TGO remains compressive (Fig. 14). This distinction in stress state, with and without a growth strain, has important implications for fatigue, as elaborated below. The decisive insights are provided by a comparison of the local displacements and strains in the substrate, with
and without TGO elongation. The current location of the crack front is especially helpful (Fig. 16a). Note that, without TGO elongation (red curve),1 the front oscillates upward and downward without a net displacement per cycle. When an elongation strain is incorporated (blue) the downward displacement is larger, resulting in a cycleby-cycle crack extension. The strains of interest are those just ahead of the TGO/crack (Fig. 16b and c). Upon reducing the applied strain, the crack front retracts as the crack opens, and the stresses ahead of the crack become tensile (Fig. 14). Introducing e_ growth has a number interrelated effects. Each time the crack is open, elements in the substrate ahead of the crack develop incremental plastic stretch epl xx normal to the crack (Fig. 16b), with a corresponding orthogonal retraction, epl yy (Fig. 16c). These effects happen because the state of in-plane compression (Fig. 14) enables e_ growth to elongate the TGO, by ‘‘pushing” it into the substrate and accommodating its displacement through lateral plastic flow, out from the crack front. A schematic that illustrates the overall effect and establishes the influence of e_ growth on TGO elongation as the fundamental basis for fatigue is presented in Fig. 17. 4.2. The creep version In this version, the substrate creep law is used with two choices for the strain rate coefficient (_e0 ¼ 4 108 and 4 109 s1 ) that encompass the measurements for
1 For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.
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Fig. 15. Contours of plastic strain in the substrate: (a) at maximum applied strain and (b) at zero applied strain.
the substrate (Table 1). The time for which the strain is imposed, tstrain, has been chosen to enable non-linearity upon straining, during the first few cycles. The ensuing stress–strain responses ascertained over 50 cycles, with 1 h hold for each, are presented in Fig. 18. Note the similarity between these predictions and the prototypical measurements presented schematically in Fig. 1b, in the sense that
the tension at zero strain increases systematically as cycling proceeds. A substantial shift in the stress at zero strain occurs after the first cycle, especially at the higher strain rate coefficient. After a few cycles, the hysteresis loop width becomes quite narrow (albeit larger for larger e_ 0 ), but large enough to cause the stress to systematically shift with each additional cycle. The hysteresis is now caused entirely by stress relaxation at the strain maximum. The cycle-by-cycle crack extension predicted by the model is presented in Fig. 19. Note that, again, there is no cycle-by-cycle crack extension when e_ growth ¼ 0 and that the extension rates are similar to those for the plasticity model for the same TGO growth rate, e_ growth ¼ 0:1% for a 1 h cycle. Moreover, da/dN increases slightly as the creep coefficient for the substrate increases. To provide additional perspective, the stress in the substrate ahead of the crack has been monitored during a strain cycle (Fig. 20). Evidently, even though the remote stresses induced by the straining deviate only slightly from elastic values, the concentrated stresses ahead of the crack become quite large (when the strain is applied) and relax quickly during the hold period. This relaxation is accompanied by strains that accommodate extension of the crack, precisely analogous to their role in the foregoing plasticity model. 5. Crack extension in the bond coat (stage I)
Fig. 16. Displacements and strains induced in the substrate throughout two strain cycles in ‘‘steady state” (cycles 10–12): a comparison of trends with and without a growth strain. (a) Displacements of the crack front. (b) The exx strains in an element of the substrate just ahead of the TGO. (c) The eyy strains in an element of the substrate just ahead of the TGO.
The intention of this section is to demonstrate that the same model can be used to replicate the extension of Vshaped fatigue cracks within the bond coat, in stage I. For this purpose, numerical calculations have been performed by incorporating the experimentally measured crack shapes and TGO thickness and implementing the plasticity version of the fatigue model. The bond coat is assigned a lower yield strength than the substrate to reflect its lower creep strength (Table 2). The crack depth is taken to be a0 = 20 lm. The included angle is taken to be at the lower end of the range observed experimentally, # = 20°. Otherwise, all of the parameters used for the stage III predictions are retained (rgrowth 300 MPa, hTGO 3 lm, rsub Y ¼ 100 MPa). Because the TGO that forms on the bond
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Fig. 17. A schematic indicating the stresses induced and the matrix mass flux governing the fatigue mechanism during one strain cycle.
coat in stage I is expected to grow more slowly than that formed on substrate-penetrating cracks, the growth rate is assigned values at the lower end of the range, 0 6 egrowth 6 0:05%=cycle. The yield strength of the bond coat, rbc Y , is varied between 10 and 40 MPa (Table 2). Superposing the bond coat does not affect the macroscopic stress–strain loops presented in Fig. 12, but it does influence the crack extension rate. A synopsis of results obtained using rgrowth and e_ growth as input variables is presented in Fig. 21. Note the almost linear trend in crack growth with the elongation strain rate and the strong trend toward reduced growth rate as the bond coat becomes stronger. It is also apparent that the growth rates are lower than those in stage III, for the same parameter choices. The reduction is attributed to several effects that have yet to be deconvoluted. (i) The included angle # = 20° diminishes
the in-plane stress induced in the TGO upon retraction during de-straining, thereby affecting its elongation. The factors that govern this angle have not yet been determined. (ii) There is a bilayer effect. Namely, since the overall strains are dictated by the strength of the substrate (rather than the bond coat), the plastic strains generated in the bond coat during cycling are much smaller than would otherwise be expected from their low strength. (iii) The e_ growth is smaller in stage I, due to the different TGO composition. 6. Implications and concluding remarks A straightforward model capable of characterizing sustained load low-cycle fatigue has been presented. It embodies the same mechanics and phenomena previously used to
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Fig. 20. The stresses induced during the de-straining and re-straining phases of the cycle, as well as the initial portion of the hold period, for the creep version of the model. The remote stress as well as the stress ahead of the crack in the superalloy are shown. The times are with reference to that for which the strain is imposed, tstrain.
Fig. 18. The evolution of stress–strain loops predicted using the creep version of the fatigue model for two values of the creep rate coefficient for the superalloy.
successfully predict rumpling of TGO on bond coats [16– 18]. It can be envisaged as a ‘‘replenishing toothpaste” model. The following steps are involved. (i) After a few strain cycles, because of the creep deformation of the superalloy, a tensile stress develops during the de-straining phase of the cycle. This stress opens cracks present in the
Fig. 19. The crack growth rates predicted in stage III upon using the creep version of the model.
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Fig. 21. (left) The extension of V-notch cracks in stage I and (right) trends in the extension with growth strain and bond coat yield strain.
material and exposes the surfaces to the atmosphere, causing oxide growth. (ii) Dilatation takes place when the alloy is converted to oxide, with associated elongation strain rate, e_ growth , that induces a compressive growth stress, rgrowth . (iii) During the re-straining phase of the cycle, the transverse extension of the substrate (required by the Poisson effect) induces in-plane (y-direction) tension in the TGO, which ‘‘pushes” the TGO into the substrate along the crack front. (iv) Concurrent creep of substrate allows plastic flow out from the crack front to accommodate the extension of the TGO. These steps occur during each strain cycle. The variables affecting the stage III crack extension rate, da/dN, include: the applied strain range, eappl; the elongation strain rate in the TGO as it grows, e_ growth ; as well as the ensuing growth stress, rgrowth ; the creep strength of the superalloy, manifest here as the reference stress, r0 ðrY Þ; the TGO thickness, hTGO; and the crack length, a. While some trends have been identified by FE simulation, because of the large number of total variables, the development of an analytical version of the model is needed before all aspects of the dependence of da/dN on material properties and test conditions can be elucidated. Such a model is currently being pursued. The same mechanics is involved in stages I and II, but the influence of the additional variables has yet to be discovered. Preliminary results have demonstrated the importance of the relative strengths of the bond coat and alloy as well as of the chemistry of the TGO, manifest in the magnitude of e_ growth These results infer that the lower da/dN in stages I and II (relative to stage III) is (at least in part) governed by the lower TGO elongation rate, e_ growth The additional effects on da/dN of the included angle and of the bilayer plastic strains remain to be elucidated. These preliminary calculations confirm that the important systems design parameters include not only the creep resistances
but also the oxidation kinetics of the coating, the IDZ and the superalloy substrate in a coupled non-linear manner. Acknowledgements Two of the authors (A.G.E. and M.Y.H.) are indebted to David Shifler and the Office of Naval Research for financial support and encouragement through Contract No. N00014-08-1-03. References [1] Evans AG, Mumm DR, Hutchinson JW, Meier GH, Pettit FS. Prog Mater Sci 2001;46:505. [2] Evans AG, Clarke DR, Levi CG. J Eur Ceram Soc 2008;28:1405–919. [3] Clarke DR. Curr Opin Solid State Mater Sci 2002;6:237. [4] Levi CG. Curr Opin Solid State Mater Sci 2004;8:77–91. [5] Isobe N, Sakurai S. Mater Sci Res Int 2003;1:29–33. [6] Marchionni M, Osinkolu GA, Onofrio G. Int J Fatigue 2002;23:1261–7. [7] Zhang JX, Harada H, Ro Y, Koizumi Y, Kobayashi T. Acta Mater 2008;56:2975–87. [8] A. Suzuki, Gen Electr Global Res, in preparation. [9] Holmes JW, McClintock FA. Metall Trans 1990;21A:1209–22. [10] Moretto P, Bressers J. J Mater Sci 1996;31:4817–29. [11] Totemeier TC, King JE. Metall Mater Trans 1996;27A:353–61. [12] Zhang YH, Knowles DM, Withers PJ. Scripta Mater 1997;37:815–20. [13] Telesman J, Ghosn LJ. Eng Fract Mech 1989;34:1183–96. [14] Crompton JS, Martin JW. Metall Trans 1984;15A:1711–9. [15] Antolovich SD, Lerch B. Cyclic deformation fatigue and fatigue crack propagation in Ni-base alloys. In: Tien JK, Caulfield T, editors. Superalloys Supercomposites and Superceramics. New York: Academic Press; 1989. p. 363–412. [16] Balint DS, Hutchinson JW. J Mech Phys Solids 2005;53:949. [17] Davis AW, Evans AG. Metall Mater Trans 2006;37A:2085–95. [18] Davis AW, Evans AG. Oxid Met 2006:1573–4889. [19] Tolpygo VK, Clarke DR. Acta Mater 2000;48:3283–93. [20] Tolpygo VK, Clarke DR. Acta Mater 2004;52:615–21. [21] Nychka JA, Clarke DR. Oxid Met 2005;63(5–6):325–52.
A.G. Evans et al. / Acta Materialia 57 (2009) 2969–2983 [22] Tolpygo VK, Clarke DR. Mater High Temp 2003;20:261–71. [23] Karlsson AM, Hutchinson JW, Evans AG. J Mech Phys Solids 2002;50:1565. [24] Reddy A, Hovis DB, Heuer AH, Paulikas AP, Veal BW. Oxid Met 2007:67. [25] Heuer AH, Reddy A, Hovis DB, Veal B, Paulikas A, Vlad A, et al. Scripta Mater 2006;54:1907–12. [26] Veal BW, Paulikas AP, Hou PY. Appl Phys Lett 2007;90:121914. [27] Birks N, Meier GH, Pettit FS. High Temperature Oxidation of Metals. Cambridge: University Press; 1988. [28] Frost HJ, Ashby MF. Deformation Mechanism Maps. New York: Pergamon Press; 1982.
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[29] Pollock TM, Field RD. Dislocations and high temperature plastic deformation of superalloy single crystals. In: Nabarro RN, Duesbery MS, editors. Dislocations in Solids, vol. 11F. Amsterdam: Elsevier; 2002. [30] Pan D, Chen MW, Wright PK, Hemker KJ. Acta Mater 2003;51:2205–17. [31] Van Sluytman JS, Suzuki A, Helmik R, Bolcavage A, Pollock TM. Superalloys 2008. Warrendale, PA: TMS; 2008. p. 499. [32] Goodall IW, Leckie FA, Ponter ARS, Townley CHA. J Eng Mater Technol 1979;101:349–55. [33] MacLachlan DW, Knowles DM. Fatigue Fract Eng Mater Struct 2001;24:503–21.
Acta Materialia 57 (2009) 2969–2983 www.elsevier.com/locate/actamat
A mechanism governing oxidation-assisted low-cycle fatigue of superalloys A.G. Evans a, M.Y. He a,*, A. Suzuki b, M. Gigliotti b, B. Hazel c, T.M. Pollock d a
Materials Department, University of California at Santa Barbara, Santa Barbara, CA 93106, USA b General Electric Global Research, Niskayuna, NY 12309, USA c General Electric Aviation, Cincinnati, OH 45215, USA d Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109, USA Received 9 December 2008; received in revised form 28 February 2009; accepted 28 February 2009 Available online 1 April 2009
Abstract A model capable of characterizing oxidation-assisted low-cycle fatigue is described. It involves the following steps. After a few strain cycles, because of creep, a tensile stress develops during the de-straining phase of the cycle. This stress opens cracks present in the material and exposes the surfaces to the atmosphere, causing thermally grown oxide (TGO) growth. Dilatation takes place upon converting the alloy to oxide, with an associated strain rate that induces a compressive growth stress. Thereafter, during the re-straining phase of the cycle, transverse extension of the substrate induces in-plane tension in the TGO, which ‘‘pushes” the TGO into the substrate along the crack front. Finite element simulations of this process have been presented that predict crack growth per cycle, da/dN, comparable with experimental measurements. Trends in da/dN with the TGO dilatation rate and the creep strength of the superalloy have been elucidated. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Low-cycle fatigue; Superalloys; Thermally grown oxide; Finite element
1. Introduction Airfoils used in gas turbines incorporate coatings for oxidation and thermal protection [1–4]. The coatings can affect the integrity of the airfoil since thermomechanical fatigue initiates at the component surface. Over the spectrum of temperatures, strain states and stresses generated, one of the most detrimental fatigue mechanisms is that upon isothermal, sustained peak low-cycle fatigue (splcf) [5–7]. This mechanism occurs at temperatures sufficient to induce creep of the superalloy, upon exposure to cyclic compressive displacements under oxidizing conditions (Fig. 1a). The creep at the peak strain partially relaxes the stress, whereupon tension develops as the displacement returns to zero. Indeed, after multiple cycles, the tensile stress and compressive stresses attain similar levels *
Corresponding author. Tel.: +1 805 893 7166; fax: +1 805 893 8486. E-mail address: [email protected] (M.Y. He).
(Fig. 1b). Actual measurements are presented elsewhere [8]. The development of tension, in conjunction with oxidation, is regarded as a primary motivation for fatigue, in accordance with the following generalities. Multiple crack-like imperfections develop from the surface of the bond coat [9–12], then propagate on a cycle-by-cycle basis, and evolve with cycling in three distinct stages, illustrated in Fig. 2. In stage I, they extend along the surface, are Vshaped in profile, and tend to arrest at the interdiffusion zone (IDZ) between the coating and the substrate. After further cycling, during stage II, the cracks enter the IDZ, and accelerate with reduced included angle. Once they reach the substrate, in stage III, some penetrate, and accelerate again, in accordance with a constant-width configuration. There is also a subsequent, stage IV, when a mixed-mode crack forms and extends along the {1 1 1} planes [12–15]. Once this happens, most of the cyclic life has been consumed, and consequently this stage is not emphasized in this paper. Throughout, a thermally grown
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.02.047
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Nomenclature a E hTGO n N S tstrain d epl e_
crack half length Young’s modulus TGO thickness creep exponent strain-hardening exponent crack spacing time period for imposing the strain during each cycle crack opening on surface plastic strain strain rate
oxide (TGO) layer forms on the crack surfaces, and exerts a central influence on the fatigue mechanism. In stage I, the TGO is largely a-Al2O3. Thereafter, refractory metal and ternary oxides form in addition to alumina, causing the oxidation kinetics to accelerate as the crack progresses downward from the surface. The intent of this paper is to examine the mechanics of cracks subject to compressive strain cycling and thereby ascertain the roles of the oxide layer as well as the constituent creep strengths on the crack extension per cycle. The paper is organized as follows. A synopsis of the salient measurements and observations of fatigue cracking is presented. Thereafter, a connection between the fatigue mechanism and TGO rumpling [16–20] is postulated, along with a synopsis of the associated phenomena. The fatigue model is presented and numerical results generated for stage III to demonstrate that the measured phenomena can be predicted without any additional mechanics and physics. The features that emerge are used to elucidate
substrate creep rate e_ sub reference strain rate for creep e_ 0 egrowth unconstrained elongation strain for TGO during growth # included angle of V-shaped crack in stages I and II rgrowth stress in TGO as it thickens reference stress for creep r0 von Mises stress req yield strength of substrate alloy rsub Y rbc yield strength for bond coat Y
the parameters that affect the crack extension per cycle. Numerical results for stage I are provided that relate fatigue to the properties of the bond coat. 2. Synopsis of measurements and observations Cylindrical test specimens of the superalloy Rene N5 (12 mm diameter) with a b-phase, aluminide bond coat were subjected to isothermal testing at 1090 °C. Axial (xdirection) compressive displacements were imposed, corresponding to a strain amplitude of 0.35%. Tests were interrupted after various fractions of the cyclic life and characterized. The evolution of the cyclic stress–strain response with cycling is presented schematically in Fig. 1. A detailed description of the stress evolutions and the crack growth process will be published elsewhere [8]. Note that, at these temperatures, the Young’s modulus is E 74 GPa (loading along [0 0 1]) and that the stress relaxation during each hold period is small relative to the total stress.
Fig. 1. Schematics of the applied straining conditions associated with sustained peak low-cycle fatigue. (a) The compressive strains imposed and trends in the stresses induced as cycling proceeds. (b) The stress–strain loops and their evolution with cycling.
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Fig. 2. Schematics of the fatigue cracks during their three stages of development. (a) V-shaped stage I cracks contained within the bond coat. (b) Stage III parallel-sided cracks within the substrate.
Multiple cracks are apparent along the surface after less than 10% of the cyclic life (Fig. 3). They have small opening displacement, d 2 lm (or less), length 2a 100 lm (or less), spaced apart by s 100 lm. After about 50% of life, the openings are much larger, d 20 lm, and several contiguous cracks have coalesced. Cross-sections (Fig. 4) reveal that, with time at temperature, the bond coat converts to two-phase b/c0 , while the IDZ becomes largely c0 . Moreover, under these test conditions, stage I persists during the first 50% of life, with the cracks invariably in the b phase with included angles of 20 6 # 6 30 (Fig. 4a). The TGO is relatively uniform, thickness hTGO 3 lm
(Fig. 5). Thereafter, up to about 70% of life, in stage II, the cracks penetrate the c0 -phase in the IDZ, and have an included angle # 15° (Fig. 4b). In stage III, the cracks propagate in the substrate with parallel sides and TGO thickness again, hTGO 3 lm (Fig. 4c). The crack growth rate, da/dN, increases dramatically at the onset of stage III (Fig. 6). 3. Connections to TGO rumpling Prior assessments of TGO rumpling upon thermal cycling have revealed crack-like imperfections in the bond
Fig. 3. A scanning electron micrograph of the top of the bond coat after about 10% of cyclic life at 1090 °C, showing stage I cracks extending along the surface normal to the strain axis.
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Fig. 4. Scanning electron images of cross-sections normal to the surface showing the fatigue cracks in profile. (a) Stage I crack in the bond coat. (b) Stage II crack that has penetrated the IDZ. (c) Stage III crack within the substrate.
coat (Fig. 7) that have features similar to stage I cracks. These features have been found experimentally [19] and replicated by numerical simulations upon finite-element
implementation of a rumpling code [18]. Because of this similarity, it is hypothesized that splcf is another manifestation of the same mechanism, albeit subject to quite dif-
Fig. 5. A higher-resolution SEM image of a stage I crack highlighting the TGO layer on its surfaces.
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Fig. 6. (a) The depth of the largest fatigue cracks as a function of the fraction of cyclic life. (b) The growth rate of the same crack, estimated by taking the derivative of a best fit through the datum points on (a).
ferent loading conditions. To explore this possibility, some of the basic phenomena involved in rumpling are summarized. The following elements are especially important. (i) During rumpling, the TGO not only thickens but also elongates. (ii) As the undulations grow, the downward displacements are accommodated by creep deformation of the bond coat.
3.1. TGO thickening and lateral straining Oxidation of the bond coat occurs by the formation of polycrystalline, columnar a-Al2O3 [2]. While most of the new oxide forms at the interface with the bond coat by inward diffusion of O, an outward counter-flux of Al causes some new a-Al2O3 to form along the through-thickness
Fig. 7. Cross-section of a crack-like perturbation caused by TGO rumpling during thermal cycling. (a) A cross-sectional image (courtesy Tolpygo and Clarke [11]). (b) Simulation conducted using a FE version of the rumpling code [9].
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Fig. 8. (a) Fractured cross-section of an alumina TGO illustrating the inner, columnar portion of the oxide formed by inward diffusion of O and the outer, equiaxed portion formed by outward diffusion of Al [2]. (b) Illustration of an alumina TGO on reoxidation after smoothly polishing the TGO formed in the first oxidation step. New oxide forms along the grain boundaries of the initially formed TGO and the amount increases with further oxidation [2].
grain boundaries, as well as at ridges along the surface [21,22] (Fig. 8). That formed on the grain boundaries must be accommodated by in-plane straining of the neighboring grains, at strain rate e_ growth [16–18,23] (Fig. 9). Hence, as shown on the left of Fig. 9, when a counter-flux of Al and O meets the internal grain boundaries with the new a-Al2O3 the neighboring grains must be compressed to accommodate the volume. Thus, e_ growth is a measure of the rate at which the TGO would elongate if theoretically detached from the bond coat and substrate. The a-Al2O3 responds to this strain rate by causing a compressive
growth stress, rgrowth (Fig. 9) [24,25]. The magnitude of this stress is governed by the creep mechanisms operating in aAl2O3 at the test temperature. This has been ascertained by measuring rgrowth in situ for the TGO as it grows on several different bond coats. This stress is of the order rgrowth 300 MPa [25] (Fig. 9, right), consistent with the deformation mechanisms for a-Al2O3 and with stress relaxation rates measured in a typical TGO [26]. When the surface is non-planar, the stress rgrowth exerts a downward pressure on the bond coat, causing it to creep [16,18], and resulting in the development of crack-like features [18].
Fig. 9. (a) When a counter-flux of Al and O meets on the internal grain boundaries to from new a-Al2O3 the neighboring grains must be compressed to accommodate the volume: e_ growth is a measure of the rate at which the TGO would elongate if theoretically detached from the bond coat and substrate. The a-Al2O3 responds to this strain rate by causing a compressive growth stress, governed by the creep mechanisms operating in a-Al2O3 at the test temperature. (b) An illustration of the stress in the TGO measured in situ in the synchrotron during a single thermal cycle from ambient to 1125 °C and back to ambient [2].
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Oxides also form on the IDZ and substrate. Their compositions differ, as already noted. The corresponding oxidation mechanisms and growth stresses have not been studied. However, these other oxides thicken more rapidly than a-Al2O3 [27], suggesting a larger elongation strain rate, e_ growth . However, they also creep more rapidly [28] (larger e_ creep ), so the resultant growth stress is indeterminate. It is treated as a parameter in the following model.
Table 2 Creep strengths of the substrate and coating layers.
3.2. Constituent creep rates
as the tension becomes substantial upon de-straining (after 10% of cyclic life, Fig. 1), the crack opens and the TGO elongates. To enable extension of the TGO-filled crack, outward creep flow must occur in the substrate around its frontal zone, similar to the flow during TGO rumpling [16,17,23]. The nature of this flow is affirmed in the following analysis [16,17]. Recall that the crack opening upon destraining is a direct consequence of the bulk creep deformation during the compressive hold, which relaxes the compressive stress. Indeed, direct evidence of uniform compressive creep throughout the sample is provided by the evolution of the precipitate morphology, with plates aligned parallel to the compression axis (Fig. 10). The principal features can be elucidated by using a plasticity model for the substrate and bond coat (with powerlaw hardening) and by invoking the reference stress method to convert to power-law creep [32]. In this method, the materials are assigned a yield strength and strain-hardening coefficient. The plastic deformations are elucidated in accordance with J2 plasticity theory, through the evolution of the von Mises stresses. The correspondence between the deformations that develop from plasticity and those induced by creep has been demonstrated for rumpling [16–18]. The viability is demonstrated below by comparing representative results based on plasticity and creep. Thereafter, for computational efficiency, trends are presented using the plasticity version. The predictions are obtained using a finite-element (FE) method: an analytical version could be pursued later, once the numerical results have delineated the phenomena. The FE code ABAQUS standard is implemented in-plane strain, with the meshes depicted in Fig. 11. The parameters are those presented in Table 1. For the plasticity analogue, the yield strength for the substrate and the strain-hardening exponents are those given in Table 1. In the simulations the crack advances through the elongation of the TGO elements in its wake, with small-scale plastic blunting occurring in the elements at the tip. Hence, the self-similarity of the crack growth in the steady state. It has not been necessary to use a crack tip rupture criterion.
Creep rates have been measured in both the substrate [29] and the bond coat [30]. In the steady state, they can be adequately represented by power-law behavior of the form: e_ s e_ 0 ðreq =r0 Þn , where req is the local von Mises stress, r0 is a reference stress, e_ 0 a reference strain rate, and n the creep exponent. Values for these parameters representative of the substrate at 1090 °C are summarized in Table 1. The corresponding creep rates for the bond coat are known [30] but are not used in this paper. The creep properties of alloys with compositions similar to those in the IDZ have been measured. They are intermediate between those of the b-NiAl bond coat and the superalloy substrate [29–31] (Table 2). 4. Substrate fatigue (stage III) The fatigue model proceeds from the rumpling model without any additional physics or mechanics, in accordance with the same numerical implementation. That is, to capture the in-plane (y-direction) straining in the TGO, in the model, e_ growth is imposed during those stages of the strain cycle when the crack is open. It is imposed at a uniform rate governed by the TGO thickness [16,17]. This strain rate causes creep of the TGO, replicated by ensuring that the von Mises stress never exceeds the growth stress, req 6 rgrowth . The ensuing response of the TGO is governed by the creep deformation of the surrounding material (bond coat, IDZ or substrate). Hence, creep deformations occurring around the tip accommodate elongation of the TGO, leading to a crack extension per cycle. The five parameters affecting the fatigue crack growth rate are thus: the TGO growth stress, rgrowth; the elongation strain rate experienced by the TGO as it grows, e_ growth ; the TGO thickness, hTGO; the creep characteristics of the surrounding material, n e_ sub e_ 0 ðr=r0 Þ ; and the crack length, a. The phenomenon is most readily visualized for stage III crack extension in the substrate (but is equally applicable to V-cracks in the bond coat and IDZ, described in Section 5). Specifically,
Composition
Creep strength (in MPa) at 1090 °C for creep rate 108 s1
Ref.
Rene´ N5 (substrate) Ni–6.3Al–15.1Pt–2Cr– 2.9Re–5.6Ta (wt.%) (IDZ) B2 (Pt,Ni)Al (bond coat)
110 60
[21] [23]
20
[22]
Table 1 Constituent material properties at 1090 °C.
TGO Substrate Bond coat
Yield strength, rY (MPa)
Young’s modulus, E (GPa)
Strain-hardening exponent, N
Strain rate coefficient, e_ 0 ðs1 Þ
Strain rate exponent, n
Creep reference stress, r0 (MPa)
300 50–200 10–40
400 74 74
– 10 10
– 4 109 –
– 10 –
– 100 –
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Fig. 10. Rene´ N5 sample with a crack extending through the IDZ into the superalloy substrate following fatigue loading with a compressive hold in the direction indicated by the arrows. Directional coarsening of the c0 precipitates with alignment parallel to the compression axis occurs due to bulk creep deformation in the superalloy during the compressive hold (Courtesy C. Brundidge).
4.1. The plasticity version The FE mesh used for the stage III calculations is depicted in Fig. 11, which shows the results after 20 cycles.
The red region is the TGO and the blue region the substrate. The horizontal region with a fine mesh represents the original location of the crack. Evidently the crack has extended to its new location after cycling. Within the
Fig. 11. (a) The FE mesh used for the stage III calculations after 20 cycles. The red region is the TGO and the blue region the substrate. The horizontal region with fine mesh represents the original location of the crack. Evidently the crack has extended to its new location upon cycling. (b) The near-tip region when the configuration is at zero strain and the crack is open. (c) The configuration when the applied strain is a maximum (crack closed except near the tip). The stage I simulations use a different mesh (not shown for brevity). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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context of the model (Fig. 11), a transverse (x-direction) cyclic displacement is imposed, at the boundaries, consistent with a compressive strain cycling between 0% and 0.4% (Fig. 1). The surface is free to displace in the y-direction. The zero displacement (dY = 0) plane is located in the substrate 5 mm from the surface. The TGO elongation strain is imposed at a constant rate, while the crack is open. This is achieved within ABAQUS by imposing an in-plane stress-free strain rate (similar conceptually to the way thermal expansion and phase changes are addressed). In the plasticity model, since the deformation is time invariant, assigning a strain/cycle (rather than a strain rate) suffices. Actual strain rates must be used in the creep model. The elongation strain range, 0 6 egrowth 6 0:1%=cycle, has been chosen because of its relevance to the prediction of rumpling rates [16–18]. In the simulations, this strain is imposed at a constant rate throughout the cycle. The growth stress is retained at, rgrowth 300 MPa [25], and the TGO thickness fixed at the measured value, hTGO 3 lm. The yield strength of the substrate is, rsub Y ¼ 100 MPa, and strain-hardening exponent, N = 10 (Table 1). All calculations are for 20 cycles with initial crack length a0 = 100 lm, such that the extension, Da, is small relative to a0. Calculations conducted for other a0 are expected to predict somewhat different da/dN: the associated trends are subject to a separate assessment. Unless stated otherwise, the results shown below are for a TGO strain, egrowth = 0.05%/cycle. The displacements presented in Fig. 11 indicate that the crack is open over a large portion of the strain cycle and, after a few cycles, remains open at the tip, throughout the entire cycle (Fig. 11c), though it closes over most of its length when fully compressed. The stresses induced at the boundaries over 20 cycles are presented in Fig. 12a. After a few cycles, steady state is reached with about equal tension and compression, with a hysteresis loop having a width, Dehyst = 0.05. The cycle-by-cycle extension of the
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crack front (Fig. 12b) reveals that, after a few cycles, the extension per cycle is invariant and (for this case), da/ dN = 26 nm/cycle, quite similar to that ascertained experimentally in stage III (da/dN = 28 nm/cycle for the corresponding crack length and similar test conditions) (Fig. 6). The profile of the crack front remains essentially unchanged as the crack opens and closes (Fig. 11). The corresponding behavior in the absence of an elongation strain rate, e_ growth ¼ 0 (Fig. 12b) confirms that continuous fatigue crack growth is linked directly to TGO formation. Additional results for other e_ growth indicate its specific influence on the crack growth rate (Fig. 13). This trend is to be interpreted from the perspective that e_ growth is composition dependent and thus affected by the distance from the surface. There is an associated influence of the substrate yield strength, with da/dN decreasing as rY increases (Fig. 13).
Fig. 13. Influence of the lateral strain rate of the TGO and of the substrate yield strength on the stage III crack extension per cycle.
Fig. 12. (a) The hysteresis loops calculated using the plasticity model. (b) Calculations of the change in crack length in stage III for a representative TGO elongation rate, e_ growth ¼ 0:1%=cycle, as well as for e_ growth ¼ 0, contrasting the two situations.
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Fig. 14. The stresses induced in the TGO during a strain cycle. (a) The in-plane stresses with and without a growth strain. (b) The concentrated rxx stresses ahead of the crack when closed and open.
Diagnosis of the stresses, strains and displacement during one cycle help to explain the phenomena governing crack extension (Fig. 14). The assessment starts with the crack closed, when the imposed compressive strain is at its maximum. Upon reducing the applied strain, the Poisson effect requires that the substrate retract along the crack plane, causing the TGO to develop in-plane compression (Fig. 14). Addition of the growth strain introduces further in-plane (y-direction) compression, causing the stress in the TGO to attain its maximum (300 MPa). Close to the front, the transverse compression in the TGO is larger, rtgo xx 600 Mpa (Fig. 14) because of the triaxiality. The corresponding stress in the substrate is also compressive, but smaller, rsub xx 120 MPa (Fig. 14). The plastic strain field has features representative of those around cracks, with the largest strains occurring within contours inclined at 30–40° to the crack plane (Fig. 15). In such [0 0 1] facecentered cubic single crystals, this inclination coincides with the orientation of the octahedral slip planes to the crack front, indicative of localized slip along these planes during fatigue [13,33]. These planes are ultimately the direction along which the cracks propagate in stage IV, when the mixed-mode stress intensity becomes sufficiently large [13]. Upon re-straining, the substrate lengthens along the crack plane, causing the TGO to develop in-plane tension (Fig. 14). The growth strain counteracts this tension such that the net in-plane stress in the TGO remains compressive (Fig. 14). This distinction in stress state, with and without a growth strain, has important implications for fatigue, as elaborated below. The decisive insights are provided by a comparison of the local displacements and strains in the substrate, with
and without TGO elongation. The current location of the crack front is especially helpful (Fig. 16a). Note that, without TGO elongation (red curve),1 the front oscillates upward and downward without a net displacement per cycle. When an elongation strain is incorporated (blue) the downward displacement is larger, resulting in a cycleby-cycle crack extension. The strains of interest are those just ahead of the TGO/crack (Fig. 16b and c). Upon reducing the applied strain, the crack front retracts as the crack opens, and the stresses ahead of the crack become tensile (Fig. 14). Introducing e_ growth has a number interrelated effects. Each time the crack is open, elements in the substrate ahead of the crack develop incremental plastic stretch epl xx normal to the crack (Fig. 16b), with a corresponding orthogonal retraction, epl yy (Fig. 16c). These effects happen because the state of in-plane compression (Fig. 14) enables e_ growth to elongate the TGO, by ‘‘pushing” it into the substrate and accommodating its displacement through lateral plastic flow, out from the crack front. A schematic that illustrates the overall effect and establishes the influence of e_ growth on TGO elongation as the fundamental basis for fatigue is presented in Fig. 17. 4.2. The creep version In this version, the substrate creep law is used with two choices for the strain rate coefficient (_e0 ¼ 4 108 and 4 109 s1 ) that encompass the measurements for
1 For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.
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Fig. 15. Contours of plastic strain in the substrate: (a) at maximum applied strain and (b) at zero applied strain.
the substrate (Table 1). The time for which the strain is imposed, tstrain, has been chosen to enable non-linearity upon straining, during the first few cycles. The ensuing stress–strain responses ascertained over 50 cycles, with 1 h hold for each, are presented in Fig. 18. Note the similarity between these predictions and the prototypical measurements presented schematically in Fig. 1b, in the sense that
the tension at zero strain increases systematically as cycling proceeds. A substantial shift in the stress at zero strain occurs after the first cycle, especially at the higher strain rate coefficient. After a few cycles, the hysteresis loop width becomes quite narrow (albeit larger for larger e_ 0 ), but large enough to cause the stress to systematically shift with each additional cycle. The hysteresis is now caused entirely by stress relaxation at the strain maximum. The cycle-by-cycle crack extension predicted by the model is presented in Fig. 19. Note that, again, there is no cycle-by-cycle crack extension when e_ growth ¼ 0 and that the extension rates are similar to those for the plasticity model for the same TGO growth rate, e_ growth ¼ 0:1% for a 1 h cycle. Moreover, da/dN increases slightly as the creep coefficient for the substrate increases. To provide additional perspective, the stress in the substrate ahead of the crack has been monitored during a strain cycle (Fig. 20). Evidently, even though the remote stresses induced by the straining deviate only slightly from elastic values, the concentrated stresses ahead of the crack become quite large (when the strain is applied) and relax quickly during the hold period. This relaxation is accompanied by strains that accommodate extension of the crack, precisely analogous to their role in the foregoing plasticity model. 5. Crack extension in the bond coat (stage I)
Fig. 16. Displacements and strains induced in the substrate throughout two strain cycles in ‘‘steady state” (cycles 10–12): a comparison of trends with and without a growth strain. (a) Displacements of the crack front. (b) The exx strains in an element of the substrate just ahead of the TGO. (c) The eyy strains in an element of the substrate just ahead of the TGO.
The intention of this section is to demonstrate that the same model can be used to replicate the extension of Vshaped fatigue cracks within the bond coat, in stage I. For this purpose, numerical calculations have been performed by incorporating the experimentally measured crack shapes and TGO thickness and implementing the plasticity version of the fatigue model. The bond coat is assigned a lower yield strength than the substrate to reflect its lower creep strength (Table 2). The crack depth is taken to be a0 = 20 lm. The included angle is taken to be at the lower end of the range observed experimentally, # = 20°. Otherwise, all of the parameters used for the stage III predictions are retained (rgrowth 300 MPa, hTGO 3 lm, rsub Y ¼ 100 MPa). Because the TGO that forms on the bond
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Fig. 17. A schematic indicating the stresses induced and the matrix mass flux governing the fatigue mechanism during one strain cycle.
coat in stage I is expected to grow more slowly than that formed on substrate-penetrating cracks, the growth rate is assigned values at the lower end of the range, 0 6 egrowth 6 0:05%=cycle. The yield strength of the bond coat, rbc Y , is varied between 10 and 40 MPa (Table 2). Superposing the bond coat does not affect the macroscopic stress–strain loops presented in Fig. 12, but it does influence the crack extension rate. A synopsis of results obtained using rgrowth and e_ growth as input variables is presented in Fig. 21. Note the almost linear trend in crack growth with the elongation strain rate and the strong trend toward reduced growth rate as the bond coat becomes stronger. It is also apparent that the growth rates are lower than those in stage III, for the same parameter choices. The reduction is attributed to several effects that have yet to be deconvoluted. (i) The included angle # = 20° diminishes
the in-plane stress induced in the TGO upon retraction during de-straining, thereby affecting its elongation. The factors that govern this angle have not yet been determined. (ii) There is a bilayer effect. Namely, since the overall strains are dictated by the strength of the substrate (rather than the bond coat), the plastic strains generated in the bond coat during cycling are much smaller than would otherwise be expected from their low strength. (iii) The e_ growth is smaller in stage I, due to the different TGO composition. 6. Implications and concluding remarks A straightforward model capable of characterizing sustained load low-cycle fatigue has been presented. It embodies the same mechanics and phenomena previously used to
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Fig. 20. The stresses induced during the de-straining and re-straining phases of the cycle, as well as the initial portion of the hold period, for the creep version of the model. The remote stress as well as the stress ahead of the crack in the superalloy are shown. The times are with reference to that for which the strain is imposed, tstrain.
Fig. 18. The evolution of stress–strain loops predicted using the creep version of the fatigue model for two values of the creep rate coefficient for the superalloy.
successfully predict rumpling of TGO on bond coats [16– 18]. It can be envisaged as a ‘‘replenishing toothpaste” model. The following steps are involved. (i) After a few strain cycles, because of the creep deformation of the superalloy, a tensile stress develops during the de-straining phase of the cycle. This stress opens cracks present in the
Fig. 19. The crack growth rates predicted in stage III upon using the creep version of the model.
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Fig. 21. (left) The extension of V-notch cracks in stage I and (right) trends in the extension with growth strain and bond coat yield strain.
material and exposes the surfaces to the atmosphere, causing oxide growth. (ii) Dilatation takes place when the alloy is converted to oxide, with associated elongation strain rate, e_ growth , that induces a compressive growth stress, rgrowth . (iii) During the re-straining phase of the cycle, the transverse extension of the substrate (required by the Poisson effect) induces in-plane (y-direction) tension in the TGO, which ‘‘pushes” the TGO into the substrate along the crack front. (iv) Concurrent creep of substrate allows plastic flow out from the crack front to accommodate the extension of the TGO. These steps occur during each strain cycle. The variables affecting the stage III crack extension rate, da/dN, include: the applied strain range, eappl; the elongation strain rate in the TGO as it grows, e_ growth ; as well as the ensuing growth stress, rgrowth ; the creep strength of the superalloy, manifest here as the reference stress, r0 ðrY Þ; the TGO thickness, hTGO; and the crack length, a. While some trends have been identified by FE simulation, because of the large number of total variables, the development of an analytical version of the model is needed before all aspects of the dependence of da/dN on material properties and test conditions can be elucidated. Such a model is currently being pursued. The same mechanics is involved in stages I and II, but the influence of the additional variables has yet to be discovered. Preliminary results have demonstrated the importance of the relative strengths of the bond coat and alloy as well as of the chemistry of the TGO, manifest in the magnitude of e_ growth These results infer that the lower da/dN in stages I and II (relative to stage III) is (at least in part) governed by the lower TGO elongation rate, e_ growth The additional effects on da/dN of the included angle and of the bilayer plastic strains remain to be elucidated. These preliminary calculations confirm that the important systems design parameters include not only the creep resistances
but also the oxidation kinetics of the coating, the IDZ and the superalloy substrate in a coupled non-linear manner. Acknowledgements Two of the authors (A.G.E. and M.Y.H.) are indebted to David Shifler and the Office of Naval Research for financial support and encouragement through Contract No. N00014-08-1-03. References [1] Evans AG, Mumm DR, Hutchinson JW, Meier GH, Pettit FS. Prog Mater Sci 2001;46:505. [2] Evans AG, Clarke DR, Levi CG. J Eur Ceram Soc 2008;28:1405–919. [3] Clarke DR. Curr Opin Solid State Mater Sci 2002;6:237. [4] Levi CG. Curr Opin Solid State Mater Sci 2004;8:77–91. [5] Isobe N, Sakurai S. Mater Sci Res Int 2003;1:29–33. [6] Marchionni M, Osinkolu GA, Onofrio G. Int J Fatigue 2002;23:1261–7. [7] Zhang JX, Harada H, Ro Y, Koizumi Y, Kobayashi T. Acta Mater 2008;56:2975–87. [8] A. Suzuki, Gen Electr Global Res, in preparation. [9] Holmes JW, McClintock FA. Metall Trans 1990;21A:1209–22. [10] Moretto P, Bressers J. J Mater Sci 1996;31:4817–29. [11] Totemeier TC, King JE. Metall Mater Trans 1996;27A:353–61. [12] Zhang YH, Knowles DM, Withers PJ. Scripta Mater 1997;37:815–20. [13] Telesman J, Ghosn LJ. Eng Fract Mech 1989;34:1183–96. [14] Crompton JS, Martin JW. Metall Trans 1984;15A:1711–9. [15] Antolovich SD, Lerch B. Cyclic deformation fatigue and fatigue crack propagation in Ni-base alloys. In: Tien JK, Caulfield T, editors. Superalloys Supercomposites and Superceramics. New York: Academic Press; 1989. p. 363–412. [16] Balint DS, Hutchinson JW. J Mech Phys Solids 2005;53:949. [17] Davis AW, Evans AG. Metall Mater Trans 2006;37A:2085–95. [18] Davis AW, Evans AG. Oxid Met 2006:1573–4889. [19] Tolpygo VK, Clarke DR. Acta Mater 2000;48:3283–93. [20] Tolpygo VK, Clarke DR. Acta Mater 2004;52:615–21. [21] Nychka JA, Clarke DR. Oxid Met 2005;63(5–6):325–52.
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