Creep Properties of an Extruded Copper–8% Chromium–4% Niobium Alloy

Materials Science and Engineering A369 (2004) 101–111

Creep properties of an extruded copper–8% chromium–4% niobium alloy

M.W. Decker, J.R. Groza, J.C. Gibeling∗

Department of Chemical Engineering and Materials Science, University of California, One Shields Avenue, Davis, CA 95616-5294, USA Received 23 June 2003; received in revised form 20 October 2003

Abstract

Constant-stress creep experiments were conducted on extruded Cu–8 Cr–4 Nb (GRCop-84) from 0.57 to 0.79 Tm, with observed strain rates spanning over four orders of magnitude. GRCop-84 is composed of approximately 14 vol.% Cr2Nb distributed in sizes ranging from approximately 30 nm to 0.5 ␮m in a matrix of pure copper with a grain size of approximately 1–3 ␮m. This high-strength alloy exhibits creep curves with a standard pure metal-type appearance and true failure strains typically in excess of 10%. The Monkman–Grant relationship is obeyed with a slope of −1.08. Computation of activation energies between neighboring isotherms yields an average value of 287 ± 14 kJ/mol. Despite apparent power law creep behavior within an isotherm, a master plot of temperature compensated strain rate (ε˙/exp[−Q/RT]) versus temperature compensated stress (σ/G) demonstrates that most of the data are above power law breakdown, consistent with σ/G>˜ 10−3.A hyperbolic sine model facilitates interpolation on this master plot to stresses and/or temperatures where experimental data do not exist. Neither the threshold stress model nor the dislocation detachment model applies to GRCop-84 under these creep conditions. © 2003 Elsevier B.V. All rights reserved.

Keywords: Power Law Breakdown; Creep; Copper; Particle-strengthened; Cu–Cr–Nb Alloy

1. Introduction Nb [1]. It is made by extruding argon atomized powders [1]. The particulate reinforcements in this alloy are Cr2Nb that Many engineering applications require the use of creep occur as approximately 30–500 nm diameter particles [2–4]. resistant materials. Of particular concern in this work is the The copper grain size is reported to be between 1 and 3 ␮m creep performance of a particle-reinforced copper alloy be- in the as-extruded material [1,3]. ing considered for the combustion chamber liner and nozzle GRCop-84 is a difficult material to categorize as being of the next generation space shuttle main engine (SSME). either dispersion strengthened (DS), precipitation hardened These components will be actively cooled, requiring high (PH), or a metal matrix composite (MMC). It is unlike tra- thermal conductivity and good creep resistance. This appli- ditional PH alloys in that the particles form during rapid cation requires a relatively pure matrix with the addition of solidification rather than by precipitating from a solid solu- non-shearable particles to improve creep strength to suitable tion. GRCop-84 is similar to a MMC in that it has a rela- levels. tively high volume fraction reinforcement with the largest One material that has been proposed for this application reinforcement size similar to the largest grain size. These is a particle-reinforced alloy developed by NASA Glenn particles are too large to interact with moving dislocations Research Center at Lewis Field (NASA GRC), known as and probably improve strength by a load transfer mecha- GRCop-84. This alloy has a fine-grained, pure copper matrix nism. The alloy is also similar to DS materials since Cr2Nb with approximately 14 vol.% particulate obtained through is also present as small (few tens of nanometers) intragran- the addition of 8 atomic percent Cr and 4 atomic percent ular particulate. The creep behavior of DS and MMC materials is poorly ∗ characterized by the power law model relationship that is Corresponding author. Tel.: +1-530-752-7037; fax: +1-530-752-9554. normally used to represent the dependence of creep strain E-mail address: [email protected] (J.C. Gibeling). rate on stress in pure metals and some metal alloys [5]. The

0921-5093/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2003.10.306 102 M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 stress exponents in particle-strengthened materials are of- [6]. This detachment stress is the mechanical stress required ten very large (20–100 [6,7]) and the activation energies are to release a dislocation from the particle/matrix interface, significantly larger than those for matrix self-diffusion [6]. after it has climbed over the particle. Since most engineer- These limitations of the power law model have led to nu- ing materials have service temperatures above zero Kelvin, merous efforts to formulate improved deformation behavior creep can occur with thermal assistance for applied stresses models in DS and MMC materials. Most of these models are less than the detachment stress. Rösler and Arzt demonstrate phenomenological in nature, but a few have a fundamental that localized climb can be stabilized if the dislocation line basis. While a single unifying model of creep behavior in energy is relaxed at the particle/matrix interface [16]. The DS and MMC materials does not exist, two prominent mod- magnitude of the detachment stress is proportional to the ex- els, the first phenomenological and the second fundamental, tent of relaxation described by the dimensionless relaxation are considered in this paper and they are discussed below. parameter k, as defined in their paper: 2 σd = σOr 1 − k (2) 2. Threshold stress model where σOr is the Orowan stress and is computed by: The basis of the threshold stress model is that the applied MGb r σOr = 0.84 √ ln (3) stress is reduced by the so-called threshold stress, resulting π(2λ − 2r) 1 − ν b in an effective stress that drives dislocations over particles. A form of this model was proposed by Barrett [8] and is where r is the average particle radius, 2λ is the interparticle used in this paper written as: spacing, M is the Taylor factor, ν is Poisson’s ratio, b is the σ − σ n magnitude of the Burgers vector, and other terms are as pre- ε˙ = A × o G (1) viously defined. The relaxation parameter can, in principal, vary from 0 to 1, but as a practical matter it is usually in the where ε˙ is the observed steady-state strain rate, A the ma- range 0.77 ≤ k ≤ 0.94 [6]. The creep rate is given by: terial constant, σ is the applied stress, σo is the threshold 2r σ 3/2 stress, G is the temperature-dependent matrix shear modu- Gb 3/2 ε˙ = ε˙o exp − (1 − k) 1 − (4) lus, and n is the stress exponent. This model has been used kBT σd to describe a number of materials regardless of whether the creep data exhibit curvature on a strain rate–stress plot, indi- where ε˙o = 6DVλρ/b is a reference strain rate, ρ is the cating the presence of a threshold (e.g., Nardone and Strife dislocation density, DV is the volume diffusion coefficient, [9]). Often investigators invoke this model as a way to re- kB is Boltzmann’s constant, and all other coefficients are as duce the apparent values of the stress exponent and acti- previously defined. In their paper, RA provide expressions vation energy to magnitudes that can be rationalized. The to compute σ/σd and k from the apparent (curve fit) stress threshold stress model has been applied to a wide variety exponent (napp), the apparent creep activation energy (Qapp), of materials [7], notably rapidly solidified aluminum alloys and the activation energy for vacancy diffusion in the pure [10], oxide DS nickel alloys [11,12], particle strengthened matrix material (Qc). intermetallics [13], and powder processed aluminum alloys The RA model has three significant shortcomings. Dislo- [14]. cation density is assumed constant and often must be made The threshold stress model has at least three notable short- unrealistically large to correlate model predictions to exper- comings. While this model does effectively describe some imental data. In addition, both the interaction parameter and creep data, it does not have a satisfactory theoretical basis. the detachment stress depend sensitively on the difference As suggested in Eq. (1), the threshold stress should depend of (Qapp − Qc) used to compute σ/σd. on temperature in proportion to (shear) modulus. In fact, The RA creep equation only applies for stresses less than some experimental results demonstrate a much stronger de- the detachment stress. As such, the detachment stress repre- pendence [7,15]. Reinforcements apparently do not cause sents an upper limit for the creep stress, whereas the thresh- the threshold stress observed in MMCs, but the threshold old stress represents a lower limit above which creep can stress scales with volume fraction of reinforcement [7,15]. occur. The detachment stress and threshold stress should not In these materials, the subtractive threshold stress may re- be equated unless a transition between these two mecha- late to the proportioning of load between the matrix and the nisms exists. particles. 3.1. Objectives

3. Dislocation detachment stress model The objective of this work is to characterize the tensile creep properties for extruded GRCop-84 tested under Rösler and Arzt (RA) have proposed a model of high tem- isothermal, isostress conditions from 773 to 1073 K. A sim- perature creep in DS materials based on a detachment stress ple power law model, as well as the two models presented in M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 103 detail above are applied to gain understanding of deforma- then cutting the “dogbone” profile, and finally stacking sev- tion mechanisms in this material. In addition, we provide a eral specimens together to drill holes in each end. A similar preliminary comparison of the creep properties of this ma- wEDM procedure has previously been used on GlidCop terial in both the rolled and extruded conditions. Al-15 [18]. A sheet of approximately 1.19 mm thick pre-production rolled GRCop-84 material was also obtained from NASA 4. Experimental methods GRC. This material has been called pre-production since it was not rolled using tension–tension methods, unlike later 4.1. Material production rolled material that was rolled using the standard tension–tension technique [17]. This material was rolled GRCop-84 is manufactured as both extruded bar stock from previously extruded bar stock followed by a 15% warm and rolled sheet (rolled from extruded material). Exami- roll per pass from 5.84 mm thick at 523 K. The process was nation of polished and etched, as-received extruded mate- completed with a 60-min stress relief anneal at 523 K. Test rial shows a non-uniform distribution of particles (Fig. 1). specimens were cut to final dimensions from the sheet using Extruded GRCop-84 exhibits regions sparsely occupied by wEDM. large particles or particle agglomerations and densely popu- Following wEDM cutting all specimens were bead blasted lated regions of small particles. The latter appear to occur in using very fine glass beads. Bead blast and recast wEDM bands parallel to the extrusion axis and the matrix grain size surface effects were partially removed from extruded spec- is smaller in these regions. The matrix grain size is larger in imen faces by dry sanding. Sanding was done in a fixture those regions with large particles or particle agglomerations. by hand on 600 grit SiC paper, which removed an average of approximately 43 ␮m in thickness and 13 ␮m along 4.2. Specimen preparation the edges of the gage section. All specimens were acetone washed prior to creep testing. Three extruded bars of GRCop-84 were obtained from NASA GRC. These bars had a steel skin surrounding a 4.3. Creep testing procedure GRCop-84 core, with a core diameter of ∼2.5 cm, and were manufactured by filling a steel can with powdered All creep experiments were conducted in tension on a GRCop-84 and extruding at ∼1133 K, with a minimum constant stress creep machine under vacuum (pressure ≤ 2× area reduction of 8:1 required for full consolidation [17]. 10−5 Torr). Constant stress was achieved through the com- Flat tensile specimens were wire electro-discharge machine bination of specimen design and a 3:1 Andrade–Chalmers (wEDM) cut on all surfaces. The specimens were fabricated (AC) cam [19]. Temperature was monitored using an by first cutting them into rectangular sheets 0.040 in. thick, Inconel-sheathed Type K thermocouple in situ. Heating was

Fig. 1. Polished and etched microstructure of extruded GRCop-84 showing both copper grains and the uneven distribution of Cr2Nb particles. Extrusion axis is horizontal. 104 M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 initiated once vacuum fell below 5×10−5 Torr. A total of 3 h Strain rate was determined for each experiment by a linear elapsed from initiation of heating to application of the mass fit to the secondary creep region. Linear fits were calculated used to begin creep testing. All specimens were furnace by a regression analysis and the linear correlation coefficient cooled post fracture (∼12 h to reach room temperature). (Pearson’s r) was maximized by adjusting which points at Displacement of the load train was monitored using a each end of the data set were included in the fit. Schaevitz 300 HR Linear Variable Differential Transformer Applied stress was scaled by the temperature-dependent coupled to a Daytronic 3130 signal conditioner. Both the shear modulus for pure copper in order to account for the conditioned LVDT signal and the millivolt level thermocou- temperature dependence of the dislocation stress fields. The ple signals were digitized via a Hewlett-Packard (HP) 3497A natural logarithm was taken of both σ/G and ε˙ before linear multiplexing multimeter resulting in a strain resolution of fits of these transformed data were conducted in Mathemat- ˜ 3.3×10−6 mm/mm and a temperature resolution of ±1.2 K. ica on a simple power model of the form ln ε˙ = A + n × ˜ Custom software was used to record data at user selectable ln σ/G. Fits to determine A and n initially included all data intervals. points within a material class (e.g., pre-production rolled or Specimens were creep tested at five nominal isotherms: extruded) and an isotherm. Subsequent fits removed a high 773 K (0.57 Tm), 823, 923, 1023, and 1073 K (0.79 Tm). This stress endpoint (possible power law breakdown) and/or a spacing was chosen to facilitate determination of creep ac- low stress endpoint (possible onset of threshold behavior). ˜ tivation energy. The results are presented in terms of the Minimizing the standard error for A and n and maximizing measured average temperatures rather than the nominal tem- the coefficient of determination (R2) maximized the qual- peratures. Applied stresses varied within each isotherm, but ity of fit. Since these parameters moved synchronously, all in general creep testing was limited at high stress by the ap- were used to determine fit quality. Power law fit results for parent onset of power law breakdown and at low stress by extruded GRCop-84 are listed in Table 1. the time to rupture. Overall, stresses ranged from 12.5 MPa The temperature dependence of creep rate can be obtained from the usual Arrhenius expression of the form: (1073 K) to 200 MPa (773 K). The longest time to failure was just less than 1600 h. Q σ n ε˙ = A × exp − × (5) RT G where Q is the activation energy, R is the universal gas con- 5. Parametric description of creep results stant, T is the temperature, and the other terms are as previously defined. Activation energy can be determined by True strain versus time plots of GRCop-84 have the ap- taking the natural logarithm of Eq. (5) and then the partial pearance of classic creep curves for pure metals with dis- derivative with respect to either T or 1/T. The resultant ex- tinct primary, secondary (steady-state), and tertiary regions pressions are: (Fig. 2). Plots of true strain rate versus true strain demon- ∂ ε˙ ε˙ strate that the secondary creep rate in extruded GRCop-84 2 ln 2 ln Q = RT ≈ RT (6a) ∂T σ T σ is constant for a range of one to two percent true strain. This G =const G =const behavior contrasts with some particle strengthened materials or that exhibit a local minimum in the strain rate versus strain ∂ ln ε˙ ln ε˙ plot (e.g., aluminum with SiC [20]). GRCop-84 is a ductile Q =−R ≈−R (6b) ∂(1/T) σ (1/T) σ material with true strain to failure often in excess of 10%. G =const G =const To compute activation energy, one must compare strain rate data from neighboring isotherms at constant σ/G. Due to 14 the high stress dependence of strain rate in GRCop-84, it 12 is difficult to generate experimental data that fully conform to this requirement. An analytical method was developed 10 instead. A range of σ/G within the power law region common 8 to two adjacent isotherms was identified and, within this 6 range, 10 equally spaced values of σ/G were identified. For σ ε˙ 4 each /G value, an interpolated value of ln was computed 764 K, 125 MPa A˜ True Strain (percent) using the power law fit values of and n for the appropriate 2 917 K, 55MPa 1068 K, 25MPa isotherm. 0 0 10 20 30 40 5.1. Extruded GRCop-84 Time (hours)

Fig. 2. Example creep curves for extruded GRCop-84. The apparent Fig. 3 is a double-logarithmic plot of strain rate versus differences in loading strain are not signiﬁcant since this quantity cannot stress for extruded GRCop-84. At a nominal temperature of be accurately measured. 1068 K, the data reveal a bilinear behavior with n = 8.5at M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 105

Table 1 Fit parameters from the power law model as applied to extruded GRCop-84 creep data Average temp (T¯ )

764 K 814 K 917 K 1012 K 1068 K 1068 K

σ/G range 2.6E−3 1.9E−3 1.1E−3 6.6E−4 4.3E−4 8.6E−4 4.4E−3 3.7E−3 2.7E−3 1.5E−3 6.0E−4 1.5E−3 ln A˜ 34.3646 33.7592 37.3814 39.2093 19.6863 46.7722 ln A˜ (S.E.) 1.508 1.785 1.358 1.580 4.386 1.414 ln A˜ (95% CI) 29.566 28.803 34.059 34.181 0.814 40.689 39.164 38.715 40.704 44.238 38.559 52.856 n 8.66 8.14 7.99 7.64 4.69 8.47 n (S.E.) 0.26 0.30 0.21 0.23 0.58 0.21 n (95% CI) 7.82 7.31 7.47 6.92 2.20 7.57 9.50 8.96 8.51 8.37 7.18 9.37 R2 0.9963 0.9934 0.9950 0.9964 0.9558 0.9982 The S.E. and 95% CI are presented for the power law parameters. high stresses and 4.7 at low stresses. Only the high-stress The average creep activation energy for extruded creep data with n = 8.5 were used in computing activation GRCop-84 is larger than the expected value of activa- energy because data in the n = 4.7 region have no com- tion energy—namely that for self-diffusion in pure cop- mon σ/G values with 1012 K data. In computing activation per (197 kJ/mol, [21]). The activation energy in extruded energies, the time weighted average temperature over the GRCop-84 is also larger than the activation energy for creep secondary creep region was used rather than the nominal in pure copper when compared at similar temperatures temperature. [22–24]. The magnitudes of the stress exponents (except The variations of activation energy (determined from for the low stresses at 1068 K) are marginally too large to Eq. (6a) and (b)) and stress exponent with temperature are directly identify a creep mechanism that is consistent with shown in Fig. 4. Averaging all activation energy values the simple power law model, but these results are typical of generates Q¯ = 286.6 ± 13.5 kJ/mol. Using the average acti- DS or MMC metals. vation energy for extruded GRCop-84, both stress and strain rate are normalized on the master plot as seen in Fig. 5. 5.2. Pre-production rolled GRCop-84

Pre-production rolled GRCop-84 creeps faster than ex- 10 -3 truded material at the same temperature and applied stress 814K n = 8.1 (Fig. 3). At 764 K, pre-production rolled GRCop-84 has a 1068K n = 8.5 917K stress exponent of 7.6 and has a creep rate approximately 10 -4 n = 8.0 10 times faster than material in the extruded condition with a stress exponent of 8.7. Very limited data exists at 917 K

) -5 10 1012K which suggests that pre-production rolled material has a -1 n = 7.6 stress exponent of 10 and has a creep rate approximately 11 times faster than extruded GRCop-84 with a stress ex- 10 -6 ponent of 8.0. The mechanisms responsible for these differ- 1068K n = 4.7 ences are being explored in a more thorough study of rolled 764K

Strain Rate (s -7 material. 10 n = 8.7

921K 5.3. Threshold stress model 10 -8 n ~ 10 765K σ σ n = 7.6 In Eq. (1), the quantity – o represents an effective stress 10 -9 or the thermal component of the stress available for creep, 10 100 200 where σo signifies the athermal component of the flow stress. Stress (MPa) Creep will only occur when the applied stress is greater than the threshold stress. Most investigators adopting this Fig. 3. Summary plot of creep properties for GRCop-84. Open symbols model to describe creep data for reinforced matrix mate- refer to extruded, closed symbols refer to pre-production rolled, and large symbols were excluded from the power law fit used to determine n rials use one of several stress exponents based on known because they represent creep rates in the power law breakdown region deformation mechanisms in the unreinforced matrix mate- (except for one flawed specimen in the 917 K isotherm). rial. Threshold stress is then determined as the intercept 106 M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111

400 10

9 350

8 300 7

250 Q (Eqn 6a) Q (Eqn 6b) 6 Stress Exponent, n n

Activation Energy, Q (kJ/mol) 200 5 700 800 900 1000 1100 Temperature (Kelvin)

Fig. 4. Variation of Q and n for extruded GRCop-84 with temperature, dotted lines show the 95% CI for n. The error bars shown on activation energy reflect the standard deviation associated in computing Q at the fit values of n and ln A˜ , but not the uncertainty in n or ln A˜ from the model fit results. on a linear–linear plot of ε˙1/n versus σ, as introduced by quently applying the Lagneborg–Bergman [25] method de- Lagneborg and Bergman [25]. scribed above. While this approach is widely utilized, it has The threshold stress model was applied to extruded and a serious deficit in that the stress exponent must be divined pre-production rolled GRCop-84, but not using the standard prior to analyzing creep data. The choice of stress exponent methods. The usual approach taken to compute the threshold naturally has an effect on the magnitudes of threshold stress stress requires first assuming a stress exponent and subse- and activation energy determined. Rather than assuming a particular stress exponent, a nonlinear fit calculated with the threshold stress model was 17 10 applied to GRCop-84 creep data using the Nonlinear Re- gression function within the Nonlinear Statistics package 16 764 K 10 814 K of Mathematica. The Levenberg–Marquardt minimization 917 K χ2 15 method was chosen to minimize the merit function. All 10 1012 K 1068 K data within an isotherm and a condition (i.e., pre-production σ 14 rolled or extruded) were fit to Eq. (1) by taking o to be 10 a temperature-dependent constant or by taking σo to scale σ 13 with the applied stress as B , where B is a real-valued, pos- 10 itive constant. In both cases, data were provided as triplets

12 of stress, shear modulus, and strain rate. Shear modulus was 10 computed using the expression given by Frost and Ashby

11 [21] at the time-weighted average temperature during sec- 10 ondary creep. Both models were evaluated as untransformed

10 data and as natural logarithm-transformed data. In the latter 10 case, the natural logarithm was taken of both strain rate and Strain Rate / exp ( - Q/RT) Strain Rate / exp ( σ 9 of Eq. (1). All models had three ﬁt parameters: A, o or B, 10 and n.

8 Proper initial values that would lead to convergence of 10 the ﬁt routine could not be found for many model per-

7 mutations. Other models generated illogical fit parameters 10 consisting of σo < 0ornThreshold Stress >nPower Law. Only 6 one attempt generated plausible fit parameters: the natural 10 -4 -3 -2 logarithm-transformed 765 K isotherm for pre-production 10 10 10 rolled GRCop-84 (Fig. 6). The stress exponent of the thresh- Stress / G(T) old stress model (5.3) is less than that of the power law Fig. 5. Master plot of steady-state creep data for extruded GRCop-84, model (7.6), as expected. Interestingly, this is the only data using Q¯ = 286.6 ± 13.5 kJ/mol. All data points are shown in this figure set among those collected that suggests the presence of a including those previously excluded from the power law fits in Fig. 3. threshold at the lowest stress (Fig. 3). The threshold stress M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 107

180 MPa -4 10 )

-1 -6 10

55 MPa -8 10

Strain Rate (s -10 10

-12 10 2 3 4 5 6 7 8 9 -3 -2 10 10 Stress / G(T)

− n Fig. 6. Threshold stress fit for pre-production rolled GRCop-84 tested at 765 K. This fit was conducted on a model of the form ε˙ = A × ((σ − σo)G 1) using 11 data points (excluding 180 MPa). Fit parameters for the threshold stress model: n = 5.3 (95% CI =±0.9), ln(A) = 19.3 (95% CI =±4.7), and σo = 26.2 (95% CI =±9.3). model may apply to rolled GRCop-84, but it does not apply were consistent with the methods used by Broyles et al. to extruded GRCop-84. [18]. The RA model was then applied to pre-production rolled and extruded GRCop-84 in the same manner as described 5.4. Rösler–Arzt model above. Where possible, constants (including υ) used in the GlidCop Al-15 analysis were used in this analysis. The vol- The RA model was applied to both extruded and ume fraction of Cr2Nb is 14.1% [1], but this number includes pre-production rolled GRCop-84 creep data using the Non- the nearly micron sized particles which are too large to inter- linear Regression function within Mathematica. All data act with dislocations and, therefore, cannot contribute to the within an isotherm were fit to a single equation using two dislocation detachment mechanism. The volume fraction of different fit schemes. In both cases, temperature-dependent small particles was estimated to be 5% (0.05) and particle shear modulus was again computed from the expression diameters of 10, 30, and 50 nm were considered. The small- given by Frost and Ashby [21]. In the first method, the est particles known to exist in GRCop-84 have a diameter ρ only fit parameter was dislocation density ( ). The second of 30–50 nm [2–4]. Fitting only dislocation density, regard- method fit the model to both dislocation density (ρ) and the less of the choice of Qc for various diffusion mechanisms activation energy for vacancy diffusion (Qc). Unfortunately, in copper (117 kJ/mol ≤ Qc ≤ 200 kJ/mol), the interaction no scheme was devised to permit a simultaneous fit on all parameter was greater than 0.94 for particle diameters of 50 five isotherms. Therefore, each isotherm was treated as a and 30 nm. Values of k>0.94 means the RA model does separate fit, even though the number of data points for these not apply because there is no detachment control. Even if isothermal fits is too small to yield good statistics. there were particles with a diameter as small as 10 nm, the This numerical method was first calibrated by applying it interaction parameter would be weak (0.921 ≤ k ≤ 0.934). to GlidCop Al-15 creep data at 773 and 973 K from Broyles Fitting both Qc and ρ either generated k>0.94 or unrea- et al. [18]. The Orowan stress, Eq. (3), interaction parame- sonable dislocation densities (ρ>1020 m−2). In summary, ter, and detachment stress, Eq. (2), were all computed based the RA model does not apply to GRCop−84 unless there on choosing Qc, napp, T, and a variety of other constants exist very small particles, but these have not been previously λ υ (including , b, , M, etc.). Poisson’s ratio was adjusted observed. to generate identical values of the Orowan stress listed in their Table 4 [18], yielding υ = 0.3. Appropriately natural logarithm-transformed isothermal pairs of data (applied 5.5. Hyperbolic sine interpolation stress and observed strain rate) were then fit to determine dislocation density. In comparing results to those reported Closer inspection of Fig. 5 suggests a transition at σ/G of by Broyles et al., interaction parameter magnitudes are du- approximately 10−3, with the limited data below this value plicated (except for one), detachment stresses are within possibly obeying a power law relationship and the data above 6%, and dislocation densities generated ranged from 9 to 2 this value exhibiting some stronger functionality. The cur- orders of magnitude lower than those reported by Broyles vature of these lower temperature and corresponding higher et al. [18]. While these results are not numerically iden- stress creep data suggest the presence of power law break- tical, they indicate that the basic methodologies employed down. Failure of the power law model at high stresses is 108 M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111

−3 17 well known, especially for σ/G ≥ 10 [21,26], although a 10 detailed microstructural understanding of power law breakdown does not exist. However, it can be phenomenologically 16 764 K 10 814 K described by a hyperbolic sine relationship of the form: 917 K 15 ε˙ σ n 10 1012 K = A sinh α (7) 1068 K D G Sinh Model 14 10 Power Law Model where D refers to diffusivity, A is a model constant, α and n ~ 4.9 13 n are both positive, dimensionless constants, and the other 10 terms have their usual meaning. The constants A and n can- 12 not be correlated to constants of the same designation from 10 other models (e.g., power law model or threshold stress 11 model). The principal utility of Eq. (7) is to facilitate inter- 10 polation to conditions for which no creep data exist. This 10 model can, of course, also be used to extrapolate outside 10

of the region where data exist, provided that no change in - Q/RT) Strain Rate / exp ( 9 operative creep mechanism occurs. 10 Eq. (7) was applied to extruded GRCop-84 creep data 8 using the Nonlinear Regression function of Mathematica. 10 Since diffusivity is included in this equation, all isotherms 7 were simultaneously fit. Data were provided either as pairs 10 of σ/G and ε/˙ exp(−Q/RT) or pairs of σ/G, ε˙ and T, where 2 6 Deff is the effective diffusivity (Deff = DL + β(σ/G) Dc, 10 -4 -3 -2 where DL is the lattice diffusion coefficient, Dc is the 10 10 10 β self-diffusion coefficient in the dislocation core, and is a Stress / G(T) constant factor of about 10) and shear modulus was computed as described above for the threshold stress model. Fig. 7. Master plot for extruded GRCop-84 creep data showing good fit − The hyperbolic sine model was evaluated as untransformed with a hyperbolic sine model above σ/G ≈ 10 3 and possible power data and as natural logarithm-transformed data. In the latter law creep below this point. All data points were included in the sinh fit (37 points) which used the average activation energy (286.6 kJ/mol). Fit case, the natural logarithm was taken of both strain rate and parameters for the hyperbolic sine model: n = 6.4 (95% CI =±0.3), of Eq. (7). A variety of fittable parameters were explored, ln(A) = 26.7 (95% CI =±1.3), and α = 379.1 (95% CI =±55.0). including A, Q (simple Arrhenius expression) or Deff , n, and α fittable or α equal to unity. A satisfactory solution to this model was obtained from a malized to create the master plot, however, the normalized natural logarithm-transformed data set. For this fit, the pre- strain rates span nine orders of magnitude and the curvature viously calculated average activation energy (286.6 kJ/mol) associated with power law breakdown is apparent. Most of was used; A, α, and n were all fittable parameters. Fig. 7 the GRCop-84 creep data lie in the power law breakdown shows this fit for extruded GRCop-84 creep data. The region because this material is extremely strong and in order six lowest σ/G data points reveal the onset of power law to obtain reasonable creep rates, the applied stresses must creep with a stress exponent of 4.9, consistent with dislo- be large. Furthermore, since most of the creep data collected cation climb controlled creep. These include the four data lie in the power law breakdown region, the threshold stress points at 1068 K that clearly showed a power law expo- and RA models are not expected to apply. nent of 4.7 (Fig. 3). Additional data at higher temperatures and/or lower stresses are needed to substantiate this idea of 5.6. Monkman-grant relationship climb-controlled power law creep below σ/G ≈ 10−3. As discussed earlier, the master plot shown in Fig. 5 was The product of minimum creep rate and rupture time developed using the activation energy computed from the in homologous units is a constant. This observation was power law model. The previous paragraph suggests that the first made by Monkman and Grant in 1956 and is repre- ε˙ = t−1 × C power law model does not apply for most of the extruded sented by the relationship, R M–G [27]. This GRCop-84 data set. This apparent disparity can be explained relationship holds for most structural materials and indi- in the following manner. Each isotherm covers less than cates that fracture is controlled by creep mechanisms. In four orders of magnitude in strain rate, which is not a large particular, the effects of stress and temperature on rupture enough range to observe the curvature caused by power law time are identical to those for creep. Fig. 8 shows that breakdown (in ε˙ versus σ space). The power law model extruded and pre-production rolled GRCop-84 follow the generates acceptable statistics due to the small region being Monkman–Grant relation quite well. Since both material modeled at any one temperature. When the data are nor- condition types are represented equally well (within some M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 109

-3 10 formation and damage are spread across the width of the 765K, E specimen and signiﬁcant surface cracking extends well -4 10 773K, R 814K, E away from the fracture surface. For large dimples, dimple )

-1 -5 917K, E diameter decreases with an increase in applied stress under 10 921K, R 1012K, E isothermal conditions or with an increase in temperature -6 10 1068K, E under isostress conditions.

-7 Some extruded GRCop-84 creep specimens show a bi- 10

Strain Rate (s modal dimple size on the fracture surface where a single -8 10 large dimple (average size approximately 5–15 ␮m) is Slope = -1.08 -9 surrounded by much smaller dimples (average size ap- 10 ␮ 2 3 4 5 6 7 proximately 0.5–1 m) located on the ridges dividing large 10 10 10 10 10 10 Rupture Time (s) dimples (Fig. 9). While the data set is small, there does not appear to be a correlation of bimodalism with stress, tem- Fig. 8. Relation between minimum creep rate and rupture time for extruded perature, strain rate, or condition (rolled versus extruded). and pre-production rolled GRCop-84. The Monkman–Grant relation is Pre-production rolled GRCop-84 exhibits small dimples, obeyed. but they are not distributed around the margins of large dimples. scatter), the fracture mechanism is the same for both rolled At creep temperatures, the fracture surface of pre-produc- and extruded material. tion rolled GRCop-84 exhibits a layered appearance at mag- niﬁcations less than approximately 3500×. These layers are parallel to the rolling plane and can be found most promi- 6. Correlation of creep properties with microstructure nently near the specimen faces (Fig. 10). Layer separation appears to be caused by the formation and growth of dimples GRCop-84 exhibits tensile ductility typically in excess of transverse to the loading axis. Some layer structures have 10% true strain at creep temperatures and in excess of 13% ribs normal to the layer, suggesting that ribs are remnant engineering strain at room temperature. Fracture surface and walls between two dimples. Dimples on the fracture sur- surface damage features of ten specimens were systemati- face are rare and large dimples are absent in pre-production cally examined to correlate the ductility and creep properties rolled GRCop-84 as compared to extruded material. noted above. Large particle-free, very smooth areas were frequently ob- These observations show that creep-tested, extruded served on the fracture surfaces in creep tested GRCop-84. GRCop-84 fails by void growth and coalescence coupled These areas are presumed to be sheared matrix material that with extensive creep crack growth and interlinkage. De- has relaxed due to diffusion and surface tension effects. As

Fig. 9. Fracture surface showing bimodal dimple size in extruded GRCop-84. Both dimple sizes are common with relatively large, particle free areas and relatively low quantities of particle debris. Extrusion and loading axes are into the page. 110 M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111

Fig. 10. Fracture surface showing layering present near specimen faces in pre-production rolled GRCop-84. Entire specimen thickness (∼1.016 mm) visible across horizontal width of photomicrograph. Rolling and loading directions are into the page; rolled sheet thickness is horizontal. noted earlier, creep experiments take several hours to cool free and show evidence of slip in the form of faint contour to room temperature, thereby providing time and tempera- lines. ture for such diffusion to occur even on the ﬁnal ligament Extensive cracking and crack bridging were observed on remaining prior to fracture. These smooth particle-free ar- the faces of all creep-tested specimens. One pre-production eas are present in both material types, but at a reduced rolled specimen revealed bridging structures in cracks 2 mm area fraction in pre-production rolled material. Some frac- from the fracture surface. The cross sectional shape of these ture surface regions in extruded material are nearly particle structures cannot be determined from the views taken, but

Fig. 11. Surface cracking 200 ␮m from the fracture surface showing crack bridging in pre-production rolled GRCop-84. Bridging ligaments appear to have polygonal cross section and may show slip bands on their faces. Rolling and loading directions are vertical; rolled sheet thickness is into the page. M.W. Decker et al. / Materials Science and Engineering A369 (2004) 101–111 111 the photomicrographs suggest that they have a complex cross References section (Fig. 11, e.g., polygonal or “+” shape). It is not known, however, if crack bridging occurs other than at the [1] D.L. Ellis, H.M. Yun. in: G.A. Eltringham, N.L. Piret, M. Sa- specimen surface. hoo (Eds.), Copper 99—Cobre 99: Fourth International Conference, The Minerals, Metals and Materials Society, Warrendale, PA, 1999, p. 417. [2] K.R. Anderson, J.R. Groza, R.L. Dreshﬁeld, D. Ellis, Metall. Mater. 7. Concluding remarks Trans. 26A (1995) 2197. [3] K.R. Anderson, High temperature coarsening of Cr2Nb precipitates The constant, true applied stress tensile creep behavior of in Cu–8 Cr–4 Nb alloy, M.S. Thesis: University of California Davis, extruded GRCop-84 has been measured from 764 to 1068 K 1996, p. 68. in vacuum. This alloy exhibits pure metal type creep curves [4] D.L. Ellis, Aerospace Structural Materials Handbook Supplement GRCop-84, National Aeronautics and Space Administration, John H. with secondary creep occurring over a one to two percent Glenn Research Center at Lewis Field: Cleveland, OH, 2001, p. 58. true strain and true failure strains typically greater than 10%. [5] A.K. Mukherjee, J.E. Bird, J.E. Dorn, ASM Trans. Quart. 62 (1969) Additionally, the Monkman–Grant relationship is obeyed for 155. both pre-production rolled and extruded GRCop-84. [6] J. Rösler, E. Arzt, Acta Metall. Mater. 38 (1990) 671. Stress exponents for extruded GRCop-84 range from 8.7 [7] J.C. Gibeling. in: R.S. Mishra, A.K. Mukherjee, K.L. Murty (Eds.), Proceedings of Creep Behavior of Advanced Materials for the 21st to 4.7, making them too large to directly apply to models Century, The Minerals, Metals and Materials Society, Warrendale, of creep deformation in pure metals. A typical Arrhenius PA, 1999, p. 239. calculation of average creep activation energy using these [8] C.R. Barrett, Trans. Metall. Soc. AIME 239 (1967) 1726. stress exponents yields an average value of 287±14 kJ/mol. [9] V.C. Nardone, J.R. Strife, Metall. Trans. 18A (1987) 109. This value is larger than the activation energy for vacancy [10] F. Carreño, O.A. Ruano, Acta Mater. 46 (1997) 159. [11] R.W. Lund, W.D. Nix, Acta Metall. 24 (1976) 469. diffusion in pure copper of 197 kJ/mol, which is a typical re- [12] J.D. Whittenberger, Metall. Trans. 15A (1984) 1753. sult for a particle-strengthened metal. This power law anal- [13] X.L. Zeng, E.J. Lavernia, Script. Metall. Mater. 32 (1995) 991. ysis has limited utility since a master plot shows power law [14] L. Kloc, S. Spigarelli, E. Cerri, E. Evangelista, T.G. Langdon, Acta breakdown in extruded GRCop-84 for σ/G>˜ 10−3. Metall. 45 (1997) 529. The threshold stress model appears to ﬁt pre-production [15] G. González-Doncel, O.D. Sherby, Acta Metall. Mater. 41 (1993) 2797. rolled creep data at 765 K. Neither the threshold stress model [16] E. Arzt, J. Rösler, Acta Metall. 36 (1988) 1053. nor the Rösler–Arzt model applies to extruded GRCop-84 [17] D. Ellis, GRCop-84 production notes, 19 March 2003: Personal creep data. These models would not be expected to apply Communication via email. for power law breakdown conditions. The successful model [18] S.E. Broyles, K.R. Anderson, J.R. Groza, J.C. Gibeling, Metall. will likely need to describe strengthening by dislocations Mater. Trans. 27A (1996) 1217. [19] E.N.d.C. Andrade, B. Chalmers, Proc. R. Soc. Lond., Series A 138 interacting with very small particles as well as a reduction in (1932) 348. stress acting on those dislocations caused by a large number [20] T.G. Nieh, K. Xia, T.G. Langdon, J. Eng. Mater. Technol. 110 (1988) of nearly grain-sized Cr2Nb particles. 77. [21] H.J. Frost, M.F. Ashby, Deformation-Mechanism Maps: the Plasticity and Creep of Metals and Ceramics. Pergamon Press, New York, Acknowledgements 1982, 166 p. [22] P. Feltham, J.D. Meakin, Acta Metall. 7 (1959) 614. [23] C.R. Barrett, O.D. Sherby, Trans. Metall. Soc. AIME 230 (1964) This research was supported by NASA Glenn Research 1322. Center at Lewis Field (NASA GRC, Cleveland, Ohio) under [24] M. Pahutová, J. Cadek,ˇ P. Rys, Philosophical Mag. 23 (1971) 509. grant NCC3-859. We would particularly like to recognize the [25] R. Lagneborg, B. Bergman, Met. Sci. 10 (1976) 20. assistance provided by Drs. David Ellis and Michael Nathal [26] W. D. Nix, J.C. Gibeling. in: R. Raj (Eds.), Flow and Fracture at Elevated Temperatures, American Society for Metals, Metals Park, of NASA GRC for their support of this project. We are also OH, 1985, p. 1. grateful to Ms. Jennifer Heelan for the metallographic and [27] F.C. Monkman, N.J. Grant, Proc. Am. Soc. Test. Mater. 56 (1956) etching protocols she developed. 593.