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Quantification of Solid Solution Strengthening and Internal Stresses metals Article Quantification of Solid Solution Strengthening and Internal Stresses through Creep Testing of Ni-Containing Single Crystals at 980 ◦C Uwe Glatzel * , Felix Schleifer , Christian Gadelmeier, Fabian Krieg, Moritz Müller, Mike Mosbacher and Rainer Völkl Metals and Alloys Department, University of Bayreuth, 95447 Bayreuth, Germany; [email protected] (F.S.); [email protected] (C.G.); [email protected] (F.K.); [email protected] (M.M.); [email protected] (M.M.); [email protected] (R.V.) * Correspondence: [email protected]; Tel.: +49-921-55-6600 Abstract: Various alloy compositions were cast as single crystals in a Bridgman vacuum induction furnace and creep tested at 980 ◦C: pure Ni, the equiatomic alloys CoCrNi and CrMnFeCoNi (Cantor alloy), single-phase fcc (Ni) solid solution alloys (with the composition of the matrix-phase of CMSX-3 and CMSX-4), and two-phase Ni-based superalloys CMSX-3 and CMSX-4. Due to the single-crystal state, grain size effects, grain boundary sliding, and grain boundary diffusion can be excluded. The results identify two major strengthening mechanisms: solid solution strengthening and other mechanisms summarized as precipitation hardening. Configurational entropy does not increase Citation: Glatzel, U.; Schleifer, F.; creep strength: The Cantor alloy, with the highest configurational entropy of all alloys tested, shows a Gadelmeier, C.; Krieg, F.; Müller, M.; weak and similar creep strength at 980 ◦C in comparison to pure Ni with zero configurational entropy. Mosbacher, M.; Völkl, R. The element Re is a very effective strengthener, both in single-phase fcc (Ni) solid solution alloys as Quantification of Solid Solution well as in two-phase superalloys. Quantitative estimations of different strengthening mechanisms: Strengthening and Internal Stresses internal back stress, misfit stresses, Orowan bowing, and g0-phase cutting (in the case of two-phase through Creep Testing of superalloys) are presented. Finite element simulations allow estimating the influence of solid solution Ni-Containing Single Crystals at 980 ◦C. Metals 2021, 11, 1130. https:// strengthening of the matrix on the creep behavior of the two-phase superalloys. doi.org/10.3390/met11071130 Keywords: superalloy; nickel alloys; creep testing; strengthening mechanisms; precipitation harden- Academic Editor: Evgeny ing; solid solution strengthening; single crystal; high entropy alloys A. Kolubaev Received: 31 May 2021 Accepted: 13 July 2021 1. Introduction Published: 16 July 2021 Single crystal Ni-based superalloys used as blade material in the hottest section of stationary gas turbines and aero-engines must withstand high static and dynamic stresses Publisher’s Note: MDPI stays neutral at high temperatures in an oxidizing atmosphere. To increase engine efficiency, the turbine with regard to jurisdictional claims in entry temperature is raised steadily, leading to ever higher material temperatures (above published maps and institutional affil- 1000 ◦C) and higher mechanical loads (several 100 MPa in tension) [1]. Due to the single- iations. crystal state and h001i orientation of the blade, these materials feature outstanding creep resistance and low elastic modulus, the latter reducing thermal fatigue stresses [1]. To further improve the creep properties of these alloys, many investigations have concentrated 0 on the optimization of microstructure, leading to cuboidal L12-ordered g precipitates Copyright: © 2021 by the authors. coherently embedded in a face-centered cubic g-matrix [2]. Others have focused on the Licensee MDPI, Basel, Switzerland. improvement of the matrix strength by alloying solid solution strengthening elements This article is an open access article such as tungsten (W), molybdenum (Mo), rhenium (Re) [3], and ruthenium (Ru) [4]. The distributed under the terms and following subchapters will discuss different contributions to the stress state present in conditions of the Creative Commons Ni-based superalloys during external creep load. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Metals 2021, 11, 1130. https://doi.org/10.3390/met11071130 https://www.mdpi.com/journal/metals Metals 2021, 11, 1130 2 of 17 Metals 2021, 11, x FOR PEER REVIEW 2 of 17 1.1. Internal Back Stress 1.1. InternalTo generally Back Stress describe the time-dependent plastic deformation at a constantly applied stressToσ generallyin the steady-state describe the regime time-dependent of creep deformation, plastic deformation the Norton at a creep constantly law: applied stress σ in the steady-state regime of creep deformation, the Norton creep law: . n # = #0·s , (1) = ∙ , . (1) with the material parameters #0 and n can be used. A variety of theories describing creep withcan the be foundmaterial in theparameters literature. If and the dislocationn can be used. climb A variety is rate-controlling, of theories describing a stress exponent creep canin thebe found range ofin 3the to literature. 7 is anticipated If the [dislocation5–9]. A stress climb exponent is rate-controlling,n = 3 is generally a stress called exponent “natural increep the range law” of [6 ].3 to 7 is anticipated [5–9]. A stress exponent n = 3 is generally called “natural creep law”Especially [6]. in multiphase alloys, stress exponents n > 10 may be observed as, for example,Especially in oxide in multiphase dispersion alloys, strengthened stress exponents (ODS) alloys n > [107]. may Such be high observed values as, of nforcan ex- be ample,explained in oxide by considering dispersion an strengthened internal back (ODS) stress alloyss0 that [7]. obstacles Such high may values exert on of dislocation n can be explainedmovement, by reducingconsidering the an applied internal stress backs [ stress8,9]: that obstacles may exert on disloca- tion movement, reducing the applied stress [8,9]: . n # = #0·(s − s0) . (2) = ∙ − . (2) When the applied stress s is in the range of the internal back stress s , the experimen- When the applied stress is in the range of the internal back stress0 , the experi- tally observed stress exponent may be found to be much higher (n > 10). This is emphasized mentally observed stress exponent may be found to be much higher (n > 10). This is em- schematically in Figure1. The green curve shows the creep rate over the external stress phasized schematically in Figure 1. The green curve shows the creep rate over the external after Equation (2) assuming n = 6 and s = 100 MPa. By application of Equation (1), for stress after Equation (2) assuming n = 6 and0 = 100 MPa. By application of Equation (1), a stress range of about 180–300 MPa, an apparent stress exponent of 12 will be measured for a stress range of about 180–300 MPa, an apparent stress exponent of 12 will be meas- (blue curve). ured (blue curve). FigureFigure 1. 1. SchematicSchematic Norton Norton plot plot of of the the stationary stationary creep creep rate rate over over the the applied applied stress stress for for a amaterial material with an internal back stress s = 100 MPa (green line) and a matrix response with n = 6 (orange line). with an internal back stress 0 = 100 MPa (green line) and a matrix response with n = 6 (orange line).For anFor applied an applied stress stress in the in rangethe range of 180–300 of 180–300 MPa, MPa, a stress a stress exponent exponent of around of around 12 is 12 determined is determined (blue . (blueline). line). The The blue blue line line is shifted is shifted vertically vertically due due to different to different#0. . TheThe center center σ swithwith respect respect to to logarithmic logarithmic scale scale is is calculated calculated by: by: n = 1 , (3) s = s0, (3) n1 − n2 with the measured high-stress exponent if the applied stress is above the internal back with n the measured high-stress exponent if the applied stress is above the internal back stress and1 the measured low-stress exponent at very high applied stress levels. At stress and n2 the measured low-stress exponent at very high applied stress levels. At stresses slightly higher than the internal back stress , the strain rate would drop down tostresses zero. This slightly will higher not happen than the since internal the internal back stress backs0 ,stress the strain is not rate a fixed would value drop but down can to locallyzero. Thisvarywill to a notlarge happen extent, since leading the internalto an always back stressnon-zero is not strain a fixed rate. value but can locally varyIn to multi-phase a large extent, alloys, leading hard to particles an always can non-zero act as dislocation strain rate. obstacles, leading to high In multi-phase alloys, hard particles can act as dislocation obstacles, leading to high internal back stresses and consequently to high , especially if the external stress σ is internal back stresses and consequently to high n1, especially if the external stress s is approaching . At very high applied stresses, the particles are no longer efficient obsta- approaching s0. At very high applied stresses, the particles are no longer efficient obstacles, cles, and therefore the stress exponent approaches the stress exponent of the material without obstacles, i.e., the matrix material. Metals 2021, 11, 1130 3 of 17 and therefore the stress exponent n2 approaches the stress exponent of the material without obstacles, i.e., the matrix material. 1.2. Solid Solution Strengthening At room temperature, the solid solution strengthening of a binary alloy can be esti- mated by the theory presented by Fleischer with a square root dependence on concentration c [10,11]: 2/3 1/2 f0 c tcb = . (4) (2T)1/2 where tc is the critical stress on a dislocation to overcome solute atoms acting as obstacles (comparable to the internal back stress s0), b is the magnitude of the Burgers vector, and f0 is the maximum interaction force between an obstacle and a dislocation.
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