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ScienceDirect Acta Materialia 61 (2013) 6506–6516 www.elsevier.com/locate/actamat

Phenomenological modeling of the eﬀect of specimen thickness on the creep response of Ni-based superalloy single crystals A. Srivastava, A. Needleman ⇑ Department of Materials Science and Engineering, University of North Texas, Denton, TX, USA Received 6 May 2013; received in revised form 28 June 2013; accepted 15 July 2013 Available online 9 August 2013

Abstract Isothermal creep tests on single-crystal Ni-based superalloy sheet specimens show a thickness-dependent creep response that is known as the thickness debit eﬀect. A size-dependent creep response at similar length scales has also been observed in a wide variety of other materials. We focus on Ni-based single-crystal superalloys and present a phenomenological nonlinear parallel spring model for uniaxial creep with springs representing the bulk and possible surface damage layers. The nonlinear spring constitutive relations model both material creep and evolving damage. The number of springs and the spring creep and damage parameters are based, as much as possible, on recent experimental observations of the thickness debit eﬀect under two creep test conditions: a low-temperature, high-stress condition, 760 °C/758 MPa, and a high-temperature, low-stress condition, 982 °C/248 MPa. The bulk damage mechanisms accounted for are the nucleation of cleavage-like cracks from pre-existing voids and, at the higher temperature, void nucleation. The surface damage mechanisms modeled at the higher temperature are an oxidation layer, a c0 -precipitate-free layer and a c0 -precipitate-reduced layer. Model results for the creep response and for the thickness debit eﬀect are in close quantitative agreement with the experimental results. In addition, the model predicts qualitative features of the failure process that are in good agreement with experimental observations. The simplicity of the model also allows parameter studies to be undertaken to explore the relative roles of bulk and surface damage as well as the relative roles of cleavage-like cracking and void nucleation in the bulk. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Creep; Creep damage; Nickel-based superalloys; Size dependence

1. Introduction Improving jet engine eﬃciency tends to drive designs towards single-crystal Ni-based superalloy turbine blades with reduced wall thickness [1,2]. The understanding of creep in single-crystal superalloy turbine blades is of importance for designing more reliable and fuel-eﬃcient aircraft engines. Current theories predict that at a given temperature the creep strain rate and the creep rupture strain should not exhibit a dependence on specimen size. However, creep tests on single-crystal Ni-based superalloys have shown increased creep strain rates and/or decreased creep rupture strains and times for thinner specimens [3–14]. This is known as the thickness debit eﬀect. ⇑ Corresponding author.

E-mail address: [email protected] (A. Needleman).

The thickness debit eﬀect occurs at size scales of the order of hundreds of microns to millimeters and at elevated temperature. This eﬀect of reducing specimen size on the creep strength at this size scale diﬀers from micron scale dislocation plasticity size eﬀects that lead to a “smaller is harder” response as in Ref. [15]. Although the focus here is on single-crystal superalloys, we note that size-dependent creep response at similar length scales has also been observed in polycrystalline superalloys (e.g. Refs. [4,16– 18]) and in a variety of other materials (e.g. Refs. [19–23]). The size scale over which the thickness debit eﬀect occurs suggests that the mechanisms leading to the thickness debit eﬀect diﬀer from those that give rise to roomtemperature dislocation plasticity size eﬀects. Indeed, the thickness debit eﬀect is generally attributed to some type of damage mechanism. One set of explanations attribute the thickness debit eﬀect to a surface damage mechanism.

1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.07.028

A. Srivastava, A. Needleman / Acta Materialia 61 (2013) 6506–6516

The basic idea is that the damage takes place in a surface layer having a specimen size-independent thickness and so is more deleterious for thinner specimens (see e.g. Refs. [3,5,24–27]). On the other hand, Baldan [7] argued that the creep response is controlled by the ratio of loss of area due to cracking to cross-sectional area so that the thickness debit eﬀect is a bulk eﬀect. Recently Srivastava et al. [13] suggested that, depending on the creep test condition, both bulk and/or surface damage mechanisms can contribute to the thickness debit eﬀect. A quantitative mechanistic model that can account for both bulk and surface damage eﬀects and that assesses their relative roles is not currently available. Here, we propose a simple phenomenological nonlinear parallel spring model for uniaxial creep with springs representing the bulk and possible surface damage layers. The nonlinear spring constitutive relations model both material creep and evolving damage. The model draws on the experimental observations of Srivastava et al. [13] and the detailed ﬁnite element calculations in Refs. [28,29]. Numerical results are presented and a quantitative comparison is made with the experimental results in Refs. [9,13] for the thickness dependence of the creep strain rate and/or the creep rupture strain and time under two loading conditions: a low-temperature, high-stress condition (760 °C/758 MPa) and a high-temperature, low-stress condition (982 °C/248 MPa). 2. Background In Srivastava et al. [13], isothermal constant nominal stress (ﬁxed force per unit initial area) uniaxial tensile creep tests were performed on sheet specimens of single-crystal PWA1484 [30], a nickel-based superalloy. The undeformed microstructure was found to contain microvoids, which had formed during the solidiﬁcation and homogenization processes, as also seen in Refs. [31,32]. The evolution of porosity depends on the stress state, as characterized by the stress triaxiality, v, and the Lode parameter, L (see Ref. [28] for calculations of porosity evolution in a single-crystal subject to creep loading). The stress triaxiality is the ratio of hydrostatic stress (positive in tension), rh, to Mises eﬀective stress, re, given by rh v¼ ð1Þ re with 1 rh ¼ ðr1 þ r2 þ r3 Þ; 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 re ¼ pﬃﬃﬃ ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr3 r1 Þ2 2

ð2Þ

The Lode parameter, L, is deﬁned as L¼

2r2 r1 r3 r1 r3

ð3Þ

Here, r1, r2 and r3 are the three principle stresses, with r1 P r2 P r3. For uniaxial tension (r2 = r3 = 0), v = 1/3

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and L = 1. In Ref. [13] it was found that, during hightemperature, low-stress creep, local changes in the nearsurface microstructure increased the local stress triaxiality to v 0.55 and the Lode parameter value to L 1. Finite element calculations of void growth in a creeping singlecrystal in Refs. [28,29] showed that void growth at these values of v and L is limited but that the evolving void shape can give a signiﬁcant stress concentration near the void surface even though creep initially relaxes stresses. In the low-temperature, high-stress condition (760 °C/ 758 MPa), no surface oxidation occurred. Furthermore, void nucleation was not observed. The only damage mechanism observed in the low-temperature, high-stress loading condition was cleavage-like cracking emanating from the void surface, as seen in Fig. 1a, at the locations of the stress concentration predicted by the ﬁnite element calculations (Fig. 1b). The extent of the cleavage-like cracks, and hence, the loss of area associated with them, was rather uniform. In the high-temperature, low-stress condition (982 °C/ 248 MPa), surface oxidation, diﬀusion and dynamic recrystallization occurred, leading to the formation of a surface region consisting of three subregions: an outermost oxide region, followed by a c0 -precipitate-free region and then a c0 -precipitate-reduced region, as shown in Fig. 2a. Signiﬁcant spallation of the surface oxide layer and cracking normal to the loading direction in the adherent oxide layer were observed, suggesting that the oxide layer carried little or no load. Extensive rafting (for more details on rafting see Refs. [33–35]) normal to the loading direction (Fig. 2a) and nucleation of new voids (Fig. 2b) were observed throughout the specimen (as also seen in Refs. [36,37]). The nucleation of cleavage-like cracks due to the stress concentration associated with void shape changes also occurred, as shown in the inset of Fig. 2b. 3. Model formulation The focus here is on modeling the damage mechanisms outlined in Section 2, which can contribute to the observed thickness debit eﬀect in the creep response of Ni-based single-crystal superalloys. We analyze a tensile specimen subjected to a ﬁxed applied force F. The deformation and strain state are assumed to be uniform along the length of the specimen, which, until near ﬁnal fracture, is consistent with the results in Ref. [13]. Hence, only a single cross-section is considered. The average nominal stress acting on the cross-section is s = F/A0, with A0 the initial cross-sectional area, and the average true stress is r = F/ A, with A the current cross-sectional area. Elasticity is ignored and the creep strain rate e_ is taken to be uniform in the cross-section. The cross-section is taken to consist of a bulk region and a surface region, as shown in Fig. 3. The current load-bearing cross-sectional area A evolves as A_ ¼ A_ bulk þ A_ surf

ð4Þ

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Fig. 1. (a) SEM image showing cleavage-like cracks emanating from a deformed void under low-temperature, high-stress creep (the loading direction is ld) [13]. (b) Finite element stress contour showing the location of the stress concentration (r11 is the local stress and R1 is the macroscopic applied stress) for a deformed void in a creeping single-crystal under uniaxial tension (along x1 axis) [28].

where Abulk = A0 and Asurf = 0 at t = 0. The evolution equations for the bulk and surface area need to be speciﬁed which strongly depend on the active damage mechanisms. The creep strain rate is given by e_ ¼ ð1 bÞC p rm þ bC s rn

ð5Þ

and the quantity b evolves as 1 b_ ¼ ð1 bÞ t0

ð6Þ

with the initial condition that b = 0 at t = 0, and t0 is a time constant that governs the transition from primary to secondary creep. The creep relations Eqs. (5) and (6) were found to give a very good ﬁt to the observed creep response in Ref. [13]. These relations can be viewed as a modiﬁcation of a creep relation given by Garofalo [38]. A power law dependence of the primary (or transient) creep rate on stress for a Ni-based single-crystal superalloy was also seen, at least approximately, in Ref. [39]. In Eq. (5), m is the primary creep exponent, n is the secondary creep exponent and Cp and Cs are parameters of the cross-section that characterize the creep resistance in primary and secondary creep, respectively. For specifying parameter values, it is convenient to write

Fig. 3. Schematic of the cross-section of the specimen showing the bulk and the surface region (surf); w and h are the initial dimensions of the cross-section.

C p ¼ e_ p

m 1 ; rp

C s ¼ e_ s

n 1 rs

ð7Þ

and to specify the values of e_ p ; rp and e_ s ; rs individually. The calculations proceed as follows: a ﬁxed tensile force F is applied and a time step Dt is chosen. At each time step, the strain increment De ¼ e_ Dt and the area changes DAbulk ¼ A_ bulk Dt and DAsurf ¼ A_ surf Dt are calculated, and the average true stress r is updated. The evolution

Fig. 2. (a) SEM image showing surface oxides (N ! Ni-rich oxide layer, M ! mixed oxide layer and A ! Al-rich oxide layer) followed by c0 -precipitatefree layer (L1) and c0 -precipitate-reduced layer (L2) under high-temperature, low-stress creep [13]. (b) SEM image showing increased number of voids under high-temperature, low-stress creep and cleavage-like cracks emanating from a deformed void is shown in the inset [13]. The loading direction is ld.

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equations for Cp and Cs diﬀer for the low-temperature, high-stress and high-temperature, low-stress cases.

a reasonably good ﬁt to the experimental data for the evolution of the creep strain.

3.1. Low temperature, high stress

3.2. High temperature, low stress

Since no surface damage was observed at 760 °C, we take A_ surf ¼ 0 so that A_ ¼ A_ bulk in Eq. (4). Following the discussion in Section 2, the damage mechanism modeled is loss of bulk area due to cleavage-like cracking emanating from the pre-existing voids. The bulk crystal material is regarded as incompressible, A_ bulk =Abulk ¼ _e, and the rate of change of load-bearing bulk cross-sectional area is written as

Surface damage due to oxidation, nucleation of new voids and cleavage-like cracking emanating from the deformed voids are active damage mechanisms in the high-temperature, low-stress loading condition. The oxide layer is taken to carry no load so that only the cross-sectional areas of the c0 -precipitate-free region, AL1, and of the c0 -precipitate-reduced region, AL2, contribute to the load-bearing surface cross-sectional area, Asurf in Eq. (4). Hence,

A_ bulk ¼ _eAbulk D_ clv

ð8Þ

where D_ clv is the rate of loss of area due to cleavage-like cracking. We assume that the nucleation of cleavage-like cracks follows Weibull statistics [40], and that a crack of ﬁxed length nucleates. The cumulative distribution function for the Weibull distribution as a function of true stress r is r k pðrÞ ¼ 1 exp ð9Þ k where k is the scale factor and k is the shape factor of the distribution. The cleavage-like cracking is assumed to begin in the secondary creep regime, which is approximately e > 0.07 [9,13]. Then, at each strain increment De ¼ e_ Dt ¼ 0:001, a random number R is generated where R 2 [0, 1] and ac e_ for R 6 p _Dclv ¼ ð10Þ 0 for R > p so that the reduction in cross-sectional area due to cleavage-like cracking for a given time step in the secondary creep regime is DDclv ¼ D_ clv Dt ¼ ac De if R 6 p or else no loss of area due to damage occurs in that increment. The parameter ac in Eq. (10) is taken such that DDclv = 0.03 mm2 per De = 0.001 if R 6 p. This corresponds to the nucleation of about 100 annular cracks extending 7 lm from the void surface as in Fig. 1a. The initial void volume fraction in the undeformed specimen cross-section corresponds to several hundred microvoids [13,36,37] and, as in Fig. 1a, multiple cracks can initiate from each microvoid. The parameter values deﬁning Cp and Cs in Eq. (7) are rp = rs = 900 MPa, e_ p ¼ 5 106 s1, m = 1, e_ s ¼ 4:5 108 s1 and n = 5. The time constant in Eq. (6) is t0 = 1.48 104 s. These values are chosen to give a good representation of the experimentally observed primary and secondary creep strain vs. time response in Refs. [9,13]. The observations in Ref. [13] do not give a direct basis for choosing the parameters in Eq. (9). Here, we take k = rp = rs, which is about 0.9 times the eﬀective yield strength of PWA1484 at 760 °C reported in Ref. [30]. For the speciﬁed value of k, k 10 gives a value of p that tends to 1 as r approaches the eﬀective yield strength and leads to

A_ surf ¼ A_ L1 þ A_ L2

ð11Þ

where AL1 = AL2 = 0 at t = 0. The depletion of the c0 -precipitate is due to surface oxidation, as noted in Ref. [13], hence the cross-sectional area of these two layers can be assumed to be proportional to the cross-sectional area of the oxide layer, Aoxide. In Ref. [13] void nucleation was seen in both the c0 -precipitate-free and c0 -precipitate-reduced regions. Hence, the load-carrying cross-sectional area of these two regions evolves as A_ L1 ¼ aL1 A_ oxide AL1 D_ void ; A_ L2 ¼ aL2 A_ oxide AL2 D_ void

ð12Þ

where aL1 and aL2 are the proportionality constants, and D_ void is the rate of homogeneous nucleation of voids. The void nucleation rate is taken to be strain controlled and to have a normal distribution about a mean nucleation strain eN as in Ref. [41]: " 2 # 1 e eN N _Dvoid ¼ D pﬃﬃﬃﬃﬃﬃ exp e_ ð13Þ 2 sN sN 2p where DN is the maximum area fraction of cross-sectional area that can be lost due to void nucleation and sN is the standard deviation. The low oxygen solubility and formation of an adherent surface oxide limits the oxidation of the superalloys to the rate of solid-state diﬀusion [42,43]. The cracks that form in the adherent oxide layer are arrested soon after their formation [13], and, in the circumstances of interest here, solid-state diﬀusion is the rate-limiting step. The oxidation kinetics, at least for the creep test conditions in Refs. [9,13], are then reasonably well approximated by a parabolic ﬁt [42,43]. The thickness x of the oxide layer at any given time t is then given by x ¼ k p t1=2

ð14Þ

The cross-sectional area Aoxide evolves as A_ oxide ¼ x_ P 0 8x_x

ð15Þ

Here, P0 = 2(w + h) is the initial perimeter of the cross-section with initial dimensions w and h (see Fig. 3).

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With the bulk crystal material regarded as incompressible, the time rate of change of the load-bearing bulk cross-sectional area is given by A_ bulk ¼ _eAbulk Abulk D_ void D_ clv

A_ oxide 1 þ aL1 þ aL2

ð16Þ

where Abulk = A0 at t ¼ 0; D_ void is given by Eq. (13), D_ clv is given by Eqs. (9) and (10), and A_ oxide is given by Eq. (15). Thus, the time rate of change of the total load-bearing cross-sectional area A is A_ ¼ A_ L1 þ A_ L2 þ A_ bulk

ð17Þ

In the high-temperature, low-stress loading condition primary creep was negligible, so we take b = 1 in Eq. (6). A power law creep relation is then taken to hold for each of AL1, AL2 and Abulk. Since each sub-area is assumed to undergo power law creep with the same creep exponent n, the overall creep relation has the form e_ ¼ C s rn

ð18Þ

where Cs is the collective creep resistance of the total loadbearing cross-section. Each sub-area is regarded as deforming homogeneously, with each planar cross-section remaining planar, so that e_ L1 ¼ e_ L2 ¼ e_ bulk ¼ e_ . The overall stress in each of the subareas is related to the strain rate by !1=n !1=n !1=n e_ e_ e_ rbulk ¼ rL1 ¼ rL2 ¼ C L1 C L2 C bulk s s s ð19Þ C bulk s

where is the creep resistance of the bulk region, C L1 s is the creep resistance of the c0 -precipitate-free region and C L2 s is the creep resistance of the c0 -precipitate-reduced region. The total force acting on the cross-sectional area is F = Abulkrbulk + AL1rL1 + AL2 rL2. The overall response is regarded as three nonlinear springs in parallel, so that average true stress r is given by Abulk AL1 AL2 rbulk þ rL1 þ rL2 ð20Þ A A A From Eqs. (18)–(20), the overall section creep resistance Cs is " #n 1 AL1 AL2 Abulk ¼

ð21Þ 1=n þ L2 1=n þ bulk 1=n Cs A C L1 A C A C r¼

s

s

s

The creep exponent n = 5 and the other parameters L2 _ L2 _ bulk deﬁning C L1 and C bulk are e_ L1 ¼ s ; Cs s s ¼ e s ¼ e s 6 1 L1 L2 1:6 10 s and rs ¼ 150 MPa, rs ¼ 300 MPa and rbulk ¼ 600 MPa, respectively. The value of C bulk was chos s sen to give a good representation of the experimentally observed creep strain vs. time response up to 1% creep L2 strain in Ref. [9]. The values of C L1 s and C s were chosen to represent a decrease in creep strength with precipitate depletion, as also presumed in Ref. [27].

The values used in the calculations for kp = 4 105 mm s1/2, aL1 = 0.5 and aL2 = 0.25 give oxide layer, c0 -precipitate-free layer and c0 -precipitate-reduced layer sizes close to those seen in Ref. [13]. Also, the values DN = 0.5, sN = 0.01 and eN = 0.012 used in Eq. (13) give an increase in porosity similar to that observed in Ref. [13]. As in the low-temperature, high-stress calculations, we take k ¼ rbulk , which is about 0.9 times the eﬀective s yield strength of the PWA1484 Ni-based single-crystal superalloy at 982 °C reported in Ref. [30], and k = 10 in Eq. (9). However, here the parameter ac in Eq. (10) is taken as DDclv = 0.01 mm2 per De = 0.001 if R 6 p because smaller length cracks were seen to nucleate in the high-temperature, low-stress creep condition in Ref. [13], as expected with the lower applied stress. 4. Results The model formulation given in Section 3 was coded in a MATLAB [44] program. As in the experiments reported in Refs. [9,13], ﬁve initial specimen thicknesses h (see Fig. 3), 0.38, 0.51, 0.76, 1.52 and 3.18 mm, are analyzed for lowtemperature, high-stress creep and four initial specimen thicknesses, h = 0.51, 0.76, 1.52 and 3.18 mm, for hightemperature, low-stress creep. The initial specimen width (w = 4.75 mm) is the same for all cases. The low-temperature, high-stress creep calculations are carried out under a constant applied nominal stress of s = 758 MPa and the high-temperature, low-stress creep calculations are carried out under s = 248 MPa, as in Refs. [9,13]. For each specimen thickness and creep condition, 50 calculations are carried out. For given conditions, the results of each calculation depends on the random number generated at each step for Eq. (10) to determine the damage due to cleavage. The average values presented subsequently are averages over the 50 calculations, and the error bars indicate the minimum and maximum values from the 50 calculations. In all calculations, the time step Dt is adjusted to give a ﬁxed strain increment De = 0.001. The time step Dt varied from 0.05 to 0.06 h in the primary and tertiary creep regimes to a maximum of 8–9 h in the steady-state creep regime. The calculations are continued until one of two conditions is met: (i) a critical strain rate is attained; or (ii) there is a critical loss of area due to cleavage-like cracking. The critical strain rate condition is taken to be e_ > e_ fail , where e_ fail is the maximum strain rate that can be sustained and the critical loss of area due to cleavage-like cracking is taken as Dclv P Dfail. In the calculations here, the value of e_ fail diﬀers for the low-temperature, high-stress and the high-temperature, low-stress conditions, while Dfail = 2 mm2 for both loading conditions. It is assumed that this much loss of load-bearing area due to crack nucleation will result in crack growth and crack coalescence, leading to fracture. There is no independent basis from the experiments in Ref. [13] for choosing e_ fail or Dfail. The

A. Srivastava, A. Needleman / Acta Materialia 61 (2013) 6506–6516 0.18

0.15

0.12

0.09

0.06

h=3.18mm (cal) h=3.18mm (avg) h=3.18mm (exp) h=0.38mm (cal) h=0.38mm (avg) h=0.38mm (exp)

0.03

0 0

200

400

600

800

t (hrs) Fig. 4. Comparison of the calculated creep curves for 50 calculations (cal) and their average (avg) with the experimental (exp) creep curves from Ref. [9] of specimens with thickness h = 3.18 and 0.38 mm for low-temperature, high-stress creep. e is the logarithmic creep strain.

values used give a reasonable quantitative ﬁt to the failure data in Ref. [13]. 4.1. Low temperature, high stress The value of e_ fail is taken to be 10_ess , where e_ ss is the creep rate taken at the beginning of steady-state/secondary creep, e = 0.07. The calculated creep strain, e, vs. time, t, curves for specimen thicknesses h = 3.18 mm and 0.38 mm are shown in Fig. 4, together with the experimental curves from Ref. [9]. The average values for both specimen thicknesses show very good correlation with the experimental curve. As seen in Fig. 4, a spread in the evolution of creep strain with time for the thinner specimens can be seen when e > 0.07. For

6511

the thicker specimens, the spread in the evolution of creep strain is small until e 0.1. The calculations for the specimens with thickness h = 3.18 and 1.52 mm are terminated after attaining Dfail for all 50 calculations, whereas the calculations for the specimens with h = 0.76, 0.51, and 0.38 mm are terminated after attaining e_ fail . Fig. 5 shows the time evolution of the Weibull cumulative distribution function, p, i.e. the probability of crack nucleation and the cumulative loss of area normalized by the initial cross-sectional area, Dclv/A0, for all specimen thicknesses h analyzed. In Fig. 5a, p starts with a small initial value of 0.3 at t 62.5 h (crack nucleation begins in the secondary creep regime). In Fig. 5b, Dclv/A0 increases with time, which raises the stress level on the remaining intact area and hence increases the probability, p, of further crack nucleation. This deleterious eﬀect is more detrimental for thinner specimens, leading to an increased fraction of the cross-sectional area being lost by cleavage-like cracks, as observed experimentally in Ref. [13]. Regardless of the criterion by which the specimen fails, the probability of crack nucleation for all specimens is near unity prior to ﬁnal failure. Fig. 6 shows the eﬀect of specimen thickness on the creep strain to fracture, ef, and on the time to fracture, tf, for the low-temperature, high-stress creep loading condition. Both the results of the model predictions and the experimental data from Refs.p[9,13] are shown. The average ﬃﬃﬃ values of ef and tf follow a 1= h dependence, as also seen in the experimental results of Srivastava et al. [13]. 4.2. High temperature, low stress In the high-temperature, low-stress creep condition, primary creep is negligible and e_ fail is taken to be 10_eN , where eN is the mean strain for void nucleation in Eq. (13).

1

0.4

0.9

0.35

h=3.18mm h=1.52mm h=0.76mm h=0.51mm h=0.38mm

0.3

0.8

Dclv /A0

0.25

p

0.7 0.6

0.2 0.15

0.5

h=3.18mm h=1.52mm h=0.76mm h=0.51mm h=0.38mm

0.4

0.1 0.05 0

0.3 0

200

400

600

800

0

200

400

t (hrs)

t (hrs)

(a)

(b)

600

800

Fig. 5. Evolution of (a) Weibull cumulative distribution function (p) and (b) cumulative loss of area due to cleavage-like cracking normalized with the initial cross-sectional area of the specimen (Dclv/A0) for low-temperature, high-stress creep for various specimens with thickness h. The values are averaged over 50 calculations.

6512

A. Srivastava, A. Needleman / Acta Materialia 61 (2013) 6506–6516 800

0.18

0.16 600

f

tf (hrs)

0.14 400

0.12 200 0.1 calculated experimental fitted curve

calculated experimental fitted curve 0

0.08 0

0.5

1

1.5

2

2.5

3

3.5

0

0.5

1

1.5

2

2.5

3

3.5

h (mm)

h (mm)

(a)

(b)

Fig. 6. The eﬀect of specimen thickness (h) on (a) the creep strain to fracture (ef) and (b) the time pﬃﬃﬃ to fracture (tf, h) for low-temperature, high-stress creep. The experimental data is taken from Refs. [9,13]. The dashed line is a least-squares ﬁt of a b= h to the mean value of the calculated ef and tf data, where a and b are positive ﬁtting constants and their values diﬀer for ef and tf. The error bars show the maximum and minimum deviation over the mean value of 50 calculations.

The calculated creep strain, e, vs. time, t, curves for specimen thicknesses h = 3.18 and 0.76 mm are shown in Fig. 7, together with the experimental results from Ref. [9]. The average values for the specimens show very good correlation with the experimental curve. In contrast to the lowtemperature, high-stress creep case, the deviation of each of the 50 calculations from the mean creep strain vs. time curve is small. All specimens failed by attaining e_ fail . The evolution of the oxide layer thickness, x, with time, t, is shown in Fig. 8a. The evolution of the c0 -precipitatefree and c0 -precipitate-reduced regions are proportional to the evolution of the oxide layer. The evolution of x depends only on time. Hence, at any given time t the thickness of the oxide layer is the same for all specimens, resulting in a relatively greater loss of load-bearing cross-sectional area for thinner specimens than for thicker ones. Also, the ﬁnal thickness of the oxide layer depends on the failure time, so that the specimens with longer creep life develop a thicker oxide scale. The extent of oxidation in Fig. 8a shows a very good correlation with the total thickness of oxide layers shown in Fig. 2a. The loss of area fraction of the bulk cross-section due to homogeneous void nucleation, Dvoid, with time, t, is shown in Fig. 8b. There is an increase in void nucleation in the bulk of the thinner specimens prior to achieving a thickness-independent saturation value of Dvoid, as shown in Fig. 8b. The dependence of Dvoid on strain e is independent of the specimen thickness since void nucleation is taken to be strain controlled (Eq. (13)). The time evolution of Weibull cumulative distribution function, p, i.e. the probability of crack nucleation is shown in Fig. 9a and the cumulative loss of area normalized with the initial cross-sectional area of the specimen, Dclv/A0, is shown in Fig. 9b. In the high-temperature, low-stress calculations the value of p is near zero until t 100 h (Fig. 9a).

The increase in r in the high-temperature, low-stress calculations depends not only on Dclv, but also on Aoxide and Dvoid. The probability of cleavage-like crack nucleation is smaller than in the low-temperature, high-stress calculations because of the smaller applied nominal stress. As shown in Fig. 9b, the loss of area fraction due to cleavage-like cracking, Dclv/A0, is slightly greater for thinner specimens than for thicker ones. Fig. 10 shows the eﬀect of specimen thickness on the creep strain to fracture, ef, and on the time to fracture, tf, for the high-temperature, low-stress creep loading condition. Both the model predictions and the experimental data from Refs. [9,13] are plotted. In the experimental data there is no systematic dependence of ef on specimen thickness h, h=3.18mm (cal) h=3.18mm (avg) h=3.18mm (exp) h=0.76mm (cal) h=0.76mm (avg) h=0.76mm (exp)

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t (hrs) Fig. 7. Comparison of the calculated creep curves for 50 calculations (cal) and their average (avg) with the experimental (exp) creep curves from Ref. [9] of specimens with thickness h = 3.18 and 0.76 mm for high-temperature, low-stress creep. e is the logarithmic creep strain.

A. Srivastava, A. Needleman / Acta Materialia 61 (2013) 6506–6516 0.5

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Fig. 8. Evolution of (a) oxide layer thickness x and (b) cumulative loss of area fraction of the bulk cross-section due to void nucleation (Dvoid) for hightemperature, low-stress creep for various specimens with thickness h. The values are averaged over 50 calculations.

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Fig. 9. Evolution of (a) Weibull cumulative distribution function (p) and (b) cumulative loss of area due to cleavage-like cracking normalized with the initial cross-sectional area of the specimen (Dclv/A0) for high-temperature, low-stress creep calculations for various specimens with thickness h. The values are averaged over 50 calculations.

whereas the calculations show some dependence of ef on h. The time to fracture, tf, shows a goodpcorrelation with the ﬃﬃﬃ experimental data and follows the 1= h dependence seen in the experimental results of Srivastava et al. [13]. Compared with the low-temperature, high-stress loading condition results in Fig. 6, there is a relative lack of scatter among the 50 computations for each specimen thickness h in Fig. 10. This is a consequence of strain-controlled void nucleation rather than cleavage-like cracking being the dominant damage mechanism in the bulk. 5. Discussion With our simple model we can consider the relative contributions of surface damage and bulk damage. Fig. 11

shows a comparison of calculated and experimental curves of creep strain vs. time for specimens with thicknesses h = 0.76 and 3.18 mm in the high-temperature, low-stress creep loading condition. In Fig. 11a only surface damage is modeled (D_ void ¼ 0 and D_ clv ¼ 0 in Eq. (16)). Surface damage alone does not lead to the rapid increase in the creep strain seen experimentally. The calculated creep curves assuming no surface damage (A_ oxide ¼ 0 in Eqs. (12) and (16)) are compared with the experimental creep curves in Fig. 11b. The evolution of creep strain for the calculations with only bulk damage also do not show the rapid increase seen in the experimental data. Hence, our model indicates that in the high-temperature, low-stress loading condition both surface and bulk damage play signiﬁcant roles. On the other hand, in the low-temperature,

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A. Srivastava, A. Needleman / Acta Materialia 61 (2013) 6506–6516 calculated experimental fitted curve

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Fig. 10. The eﬀect of specimen thickness (h) on (a) the creep strain to fracture (ef) and (b) the pﬃﬃﬃtime to fracture (tf, h) for high-temperature, low-stress creep. Experimental data is taken from Refs. [9,13]. The dashed line is a least-squares ﬁt of a b= h to the mean value of calculated ef and tf data, where a and b are positive ﬁtting constants and their values diﬀer for ef and tf. The error bars show the maximum and minimum deviation over the mean value of 50 calculations.

0.18

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h=0.76mm (exp) h=0.76mm (cal) h=3.18mm (exp) h=3.18mm (cal)

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Fig. 11. Comparison of the calculated (cal) creep curves (a) considering only surface damage ðD_ void ¼ D_ clv ¼ 0Þ and (b) considering only bulk damage ðA_ oxide ¼ 0Þ with the experimental (exp) creep curves from Ref. [9] of specimens with thickness h = 3.18 mm and 0.76 mm in the high-temperature, lowstress loading condition.

high-stress loading condition bulk damage alone can give a rather good representation of the evolution of creep strain. Our results indicate that, much like Baldan [7] suggested, the loss of load-carrying area due to bulk damage is a major contributor to the thickness debit eﬀect. Previous models of the thickness debit eﬀect in the hightemperature, low-stress loading condition focused on a surface damage mechanism [25–27]. Although surface damage alone can reproduce the thickness debit eﬀect, the surface damage layer thickness that needs to be assumed is much greater than that observed. For example, in Gullickson et al. [25], a good representation of the experimental creep curve for a specimen with h = 0.38 mm was obtained with only surface damage with a surface layer thickness of

152 lm, which is more than three times the thickness, x(1 + aL1 + aL2), of the surface damage layer in the experiments (Fig. 2a) or in our calculations (Fig. 8). Our simple phenomenological model gives a quite good prediction of the evolution of creep strain with time for various specimen thicknesses under both the creep test conditions. However, the quantitative agreement of strain to failure and time to failure with experiment depends to some extent on the more or less arbitrary choice of the values of Dfail and e_ fail . In most of the calculations here, failure is associated with attaining e_ fail . Of course, physically attaining a certain strain rate does not imply the loss of load-carrying ability. The presumption in using such a criterion is that the strain rate increase is due to the loss of

A. Srivastava, A. Needleman / Acta Materialia 61 (2013) 6506–6516

load-carrying area and that a large strain rate corresponds to a high density of defects that will interact and coalesce – a process not accounted for in the simple model. Both the experiments and the calculations exhibit a square root dependence on specimen thickness (Figs. 6 and 10). The ﬁt for the calculations is even better than for the experimental data. Nevertheless, we do not see a pﬃﬃﬃ mechanistic reason for the 1= h dependence for both the pﬃﬃﬃ strain to failure and the time to failure. Indeed, the 1= h dependence ﬁts the results for the two thicknesses, h = 1.52 mm and h = 3.18 mm, in the low-temperature, high-stress loading condition, where the failure is associated with a loss of area due to cracking, as well as for all the remaining calculations under both loading conditions, where failure is associated with the strain rate reaching e_ fail . In addition to the ﬁt of the creep curve and creep failure data, the model predicts some features of the failure process that are in very good qualitative agreement with the experimental observations, such as the increased fraction of cleavage-like cracks on the failure surfaces of thinner specimens and the thickness of the surface damage layer in the high-temperature, low-stress loading condition. The model also points out the important role played by void nucleation in the high-temperature, low-stress loading condition. Indeed, calculations (not shown here) for the high-temperature, low-stress loading conditions with no cleavage-like crack formation give only a small change in the creep strain evolution from that in Fig. 9. On the other hand, calculations for this loading condition (also not shown here) that ignore void nucleation but account for cleavage-like cracking give an evolution of creep strain that is very similar to the results computed for surface damage only in Fig. 11a. The failure mechanism that emerges from our model is a damage percolation process similar to that in Ref. [45] except that in the circumstances here the process is nucleation dominated. Depending on the temperature and stress level, the main damage mechanism is either the nucleation of cleavage-like cracks or void nucleation. At the low value of stress triaxiality in uniaxial tension, void growth does not contribute [28,29]. The surface damage under high temperature and low stress in the experiments of Refs. [9,13] plays a secondary role and exaggerates the bulk damage in thinner specimens. The loss of area associated with defect nucleation leads to increased stress and an increased strain rate on the remaining intact area, which then leads to increased defect nucleation, further reducing the load-carrying area and so on. This mechanism eventually leads to a cascade of defect nucleation that is more detrimental for thinner specimens than for thicker ones, so giving rise to the thickness debit eﬀect. 6. Conclusions We have developed a simple phenomenological model to analyze the eﬀect of specimen thickness on the uniaxial tensile creep response of Ni-based single-crystal superalloys.

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The main features of the model draw on the experimental observations in Ref. [13] and the ﬁnite element calculations of void evolution in Refs. [28,29]. However, the model contains failure parameters that were chosen to ﬁt experimental data. Although the quantitative ﬁt to the experimental data depends on the speciﬁc value of these parameters, key qualitative features do not. In the low-temperature, high-stress loading condition, the main contribution to the thickness debit eﬀect comes from bulk damage due to the nucleation of cleavage-like cracks from the surface of pre-existing voids. In the high-temperature, low-stress loading condition, bulk damage, mainly void nucleation, and surface damage due to oxidation contribute to the thickness debit eﬀect. A thickness debit eﬀect, in good quantitative agreement with the experimental data in Refs. [9,13], is obtained under both the low-temperature, high-stress and hightemperature, low-stress loading conditions. The model also predicts several features of the failure process in very good qualitative agreement p with ﬃﬃﬃ the experimental observations, such as the a b= h ﬁt to failure times and strains, and the increased area fraction of cleavage for thinner specimens. Parameter studies indicate that void nucleation plays a major role in creep failure in the high-temperature, low-stress loading condition.

Acknowledgement We are grateful for the ﬁnancial support provided by the Air Force Research Laboratory (AFRL/RXLM) to the University of North Texas under the Institute for Science and Engineering Simulation (ISES) Contract FA8650-08C-5226. We are also grateful to Professor J.R. Rice of Harvard University for insightful questioning that prompted this work. References [1] Gell M, Duhl DN, Giamei AF. In: Superalloys 1980, proceedings of the 4th international symposium on superalloys; 1980. p. 205–14. [2] Koﬀ BL. J Propulsion Power 2004;20:577595. [3] Doner M, Heckler JA. SAE technical paper 851785, Society of Automative Engineers Inc.; 1985. [4] Duhl DN. In: Sims CT, Stoloﬀ NS, Hagel WC, editors. Superalloys II. New York: John Wiley; 1987. p. 189–214. [5] Doner M, Heckler JA. In: Superalloys 1988, proceedings of the 6th international symposium on superalloys. Warrendale, PA: TMS; 1988. p. 653–62. [6] Duhl DN. In: Tien JK, Caulﬁeld T, editors. Superalloys. Supercomposites and superceramics. New York: Academic Press; 1989. p. 149–82. [7] Baldan A. J Mater Sci 1995;30:6288–98. [8] Henderson PJ. In: Earthman JC, Mohamed FA, editors. Creep and fracture of engineering materials and structures; 1997. p. 697–706. [9] Seetharaman V, Cetel AD. In: Green KA, Pollock TM, Harada H, editors. Superalloys 2004. TMS; 2004. p. 207–14.

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