Comparison of Isothermal and Adiabatic Elasticity Characteristics of the Single Crystal Nickel-Based Superalloy CMSX-4 in the Temperature Range Between Room Temperature and 1300°C

Based on the results of performed thermophysical measurements and available experimental data for adiabatic elastic stiffnesses, isothermal elastic characteristics of the single crystal nickel-based superalloy CMSX-4 have been calculated for a wide temperature range, from room temperature to 1300°C. According to the results obtained, the adiabatic and isothermal values of such elastic characteristics as elastic stiffnesses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{c}_{{11}}}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{c}_{{12}}}$$\end{document} and bulk modulus of elasticity B differ significantly at high temperatures. The reasons for this are significant changes in the thermophysical properties of the alloy with temperature and a temperature increase of its Poisson’s ratio approaching the limiting value for cubic crystals, equal to 0.5. It is shown that the use of adiabatic elastic constants instead of isothermal ones in engineering calculations affects the relationship between the volumetric strain and hydrostatic stress, and this effect is similar to introducing a field of thermal dilatation into the analyzed object. At low temperatures, this effect is small, but at high temperatures, typical for the service conditions of the blade material of aircraft gas turbine engines, it increases many times.


INTRODUCTION
The measurement of the elastic properties of materials is an important practical task, since the values of elasticity characteristics are required in many engineering calculations.Methods for measuring elastic properties are divided into two groups: static and dynamic, as described and discussed by Huntington [1].In static methods, the material is loaded either with a constant load or at a slow loading rate and the isothermal elastic compliances or stiffnesses are determined from the specimen strain, here the index T stands for temperature.The main dynamic methods are resonance and acoustic ones.In the first method, various high-frequency vibrations (flexural, torsional) are induced in the specimen and the elastic constants are calculated from the measured resonance frequencies.In the second method, ultrasonic pulses are passed through the specimen and the elastic constants are determined from the magnitude of the velocities of elastic waves.In dynamic methods, the adiabatic elastic constants or are measured, here S stands for entropy.Isothermal elastic constants are required in engineering calculations, but static methods for their measurement have low accuracy.Therefore, in practice, adiabatic elastic constants are usually measured by more precise dynamical methods and their values are used in calculations instead of isothermal ones.At low temperatures, the values of isothermal and adiabatic elastic constants differ insignificantly and, therefore, the use of adiabatic elastic constants instead of isothermal elastic constants is usually reasonable.However, in the structural analysis of mechanisms and their components with a low safety factor, such as aircraft applications, higher requirements are applied to the accuracy of the material characteristics used, including elastic constants.The difference between isothermal and adiabatic elastic constants increases with increasing temperature, which is relevant for high temperature applications.
Therefore, for such applications, either a proof of the validity of using adiabatic elastic constants instead of isothermal ones or an adequate correction of their values is required.
In the present work, the elastic characteristics of single crystal nickel-based superalloys [2][3][4] used for casting aircraft gas turbine engine blades are considered, using superalloy CMSX-4 as an example.The temperature dependence of the elastic constants of this alloy has been measured in several investigations, but usually by dynamic methods, e.g., by the resonance method in [5][6][7] and be the ultrasonic method in [8].That is, the adiabatic elastic constants were measured in these works.For aircraft gas turbine blades, the structural analysis is performed for a wide range of service temperatures, which reach 1100-1150°C in normal operation modes, but can reach 1200-1250°C in accidental situations [9].Modeling of such a technological process as hot isostatic pressing of turbine blades is carried out at even higher temperatures close to solidus, near 1300°C [10].Isothermal elastic constants are required in all these calculations.Therefore, the aim of the present work was to evaluate the difference between adiabatic and isothermal elastic characteristics of superalloy CMSX-4 over a wide temperature range and to consider the possible consequences of using experimental adiabatic elastic constants instead of the required isothermal ones in engineering calculations.

MATERIALS AND EXPERIMENTAL METHODS
The objects of this investigation were single crystals of nickel-based superalloys CMSX-4 [11] and CMSX-10 [12] (Cannon-Muskegon).The alloy CMSX-4 contains 3 wt % rhenium, and the alloy CMSX-10 contains 6 wt % rhenium.Therefore, according to the internationally accepted classification, these alloys respectively belong to the 2nd and 3rd generations of single crystal nickel-based superalloys.Cylindrical single crystals of the alloys, about 18 mm in diameter and about 170 mm in length, with an axial crystallographic orientation of [001], were solidified by the Bridgman-Stockbarger method at Howmet Alcoa.The single crystal structure of this orientation was obtained using a helicoidal-shaped grain selector, which provides selection of a single grain with orientation close to [001] from the many grains nucleated at the bottom of the starting block.A detailed investigation of this process is presented in [13].After solidification, the single crystals were subjected to a standard heat treatment including a long multi-stage homogenization annealing at temperatures between γ′-solvus and solidus, see [14], followed by two-stage aging at temperatures of 1140 °C and 870 °C.After heat treatment, the alloys had a regular γ/γ′-microstructure, in which γ-matrix, complex-alloyed solid solution of nickel (cubic structure A1), is strengthened by finely dispersed cuboidal γ′-precipitates (phase based on Ni 3 Al intermetallic, cubic structure L1 2 ), with a volume fraction of about 75%.In order to calculate the isothermal elastic constants of a material from the values of its adiabatic elastic constants, one needs to know the material density , coefficient of linear thermal expansion (CLTE), and specific heat capacity c p (T) (at constant pressure) at the temperature T under consideration.Therefore, in the experimental part of this work, the temperature dependences of thermal expansion and specific heat capacity were measured for the investigated materials in the temperature range between room temperature and 1300°C.The CLTE of single crystals was measured in a L75/240 RT high-temperature dilatometer (Linseis).For dilatometric analysis, rectangular bars were first cut longitudinally from [001] oriented single crystals, from which cylindrical samples with a diameter of 4 mm and a length of 35 mm were machined.The measurements were carried out under vacuum, and the elongation of the samples was registered by the displacement of a corundum pusher pressed against the sample with a force of about 1 N.In the dilatometer, the samples were heated in a resistance furnace at a rate of 5°C/min, and the temperature was controlled using an Stype thermocouple (Pt/Pt-10Rh) with an accuracy of ±3°C.The dilatometric system was pre-calibrated by zero measurements and a comparative measurement using a reference sample.The measurement error (standard deviation) for repeated measurements was less than 5×10 -8 °C-1 .
Calorimetric measurements were performed on disc-shaped samples with a diameter of 10 mm, thickness of about 1 mm and mass of about 0.8 g.Heat capacity was measured by laser flash method under vacuum in a TC-3000H/L thermal analyzer (SINKU-RIKO).Similar as in the dilatometric analysis, the temperature was controlled using an S-type thermocouple with an accuracy of ±3°C.A more detailed description of the method of heat capacity measurement by laser flash method in the TC-3000H/L thermal analyzer is given in [15].
The measured temperature dependences and were approximated by the equation proposed in [6] (1) where T is the temperature in °C, and a, b, c and d are fitted coefficients.The approximation was performed by the Levenberg-Marquardt algorithm using the program SciDAVis [16].

EXPERIMENTAL RESULTS
Figure 1a shows the results of dilatometric analysis of CMSX-4 and CMSX-10 single crystals.Here solid curves are experimental data, dashed curves are approximation by Eq. (1).The fitted coefficients of Eq. (1) for are presented in Table 1.It can be seen that the dilatometric curves of the investigated alloys have the same nonlinear shape, which is well described by Eq. (1).At temperatures below 600°C, the dilatometric curves of CMSX-4 and CMSX-10 are almost coincident, but at higher temperatures, CMSX-10 shows slightly larger thermal expansion than CMSX-4.
In order to obtain a temperature dependence of the CLTE , the experimental dependence has ещ be differentiated by T. However, despite the visually smooth appearance of the experimental curves , their direct differentiation is difficult due to the invisible high-frequency noise.Therefore, in the present work we did not differentiate the experimental curves directly, but their approximating function (1).
The curves plotted by Eq. ( 2) are presented in Fig. 1b.It can be seen that when the temperature increases from 20 °C to 1300 °C, the value of α increases by a factor of 2.9 for CMSX-4 and by a factor of 3.3 for CMSX-10.
It should be noted that the nonlinearity of the curves and is mainly due to the dissolution of the strengthening phase γ′ with increasing temperature, which has both a smaller lattice parameter, , as well as a smaller CLTE , as compared to those of the γ-matrix, see respectively [17] and [18].
The temperature dependence of density is calculated as where is density of CMSX-4 or CMSX-10 at room temperature, respectively equal for these alloys to 8.7 g/cm 3 and 9.05 g/cm 3 [3].From Fig. 1c follows that when the temperature increases from 20°C to 1300 °C, the densities of CMSX-4 and CMSX-10 decrease by 6.5% and 7%, respectively.
Figure 1d shows temperature dependences of specific heat capacity of CMSX-4 and CMSX-10 at constant pressure.Here dots represent experimental values, curves are approximations by Eq. ( 1).The fitted coefficients of Eq. ( 1) for the function are given in Table 1.It can be seen in Fig. 1d that the heat capacity of the alloys c p increases with increasing temperature.For CMSX-4 the value of c p at 1300°C is ≈2.5 times larger than at 20°C, while for CMSX-10 this ratio is approximately equal to 3. Similar as for the thermal expansion, the increase of specific heat capacity with temperature is mainly due to the dissolution of γ′-phase, which specific heat capacity is lower as compared to that of γ-phase [15].

CALCULATIONS AND ANALYSIS
It follows from thermodynamic considerations, see [1,19], that the values of isothermal elastic compliances of a crystal of arbitrary symmetry can be calculated from the values of its adiabatic compliances as (4) where T is the absolute temperature in degrees Kelvin, C p is the specific (per unit volume) heat capacity at constant pressure, and is the i-th component of the matrix (6 × 1) of thermal strain coefficients.For most types of crystals the angular thermal strain is zero, that is, when , and for cubic crystals the equality holds as well, where α is the isotropic CLTE.The heat capacity C p can be expressed through the specific heat capacity per unit mass c p and density ρ as .Thus, for the elastic compliances and of cubic crystals, the relationship (4) can be written as (5) where (6) The isothermal and adiabatic shear elastic compliances are equal, , as follows from (4) at .Since ∆s > 0, the inequality holds for positive elastic compliance s 11 , while the opposite is true for negative elastic compliance s 12 , .
From a physical point of view, the difference between isothermal and adiabatic elastic constants can be interpreted as follows.According to the Le Chatelier-Brown thermodynamic principle, the adiabatic elastic expansion of bodies has to be accompanied by a "compensating" process, i.e., a decrease in their temperature.Therefore, when elastic properties are measured under adiabatic conditions, tensile loading T S s s causes a drop in the temperature of the sample (similar to the Joule-Thomson effect) and consequently its thermal contraction.Therefore, the total strain measured in the loading direction under adiabatic conditions is smaller than that measured under isothermal conditions, and consequently, .In contrast, the transverse compressive strain is larger than that measured under isothermal conditions, and thus .The shear strain is not affected by isotropic thermal dilatation and, therefore, .
Figure 2 shows a graph of the temperature dependence of the difference between isothermal and adiabatic elastic compliances Δs for CMSX-4 and CMSX-10 plotted using formula (6) and temperature dependences of thermophysical characteristics of these alloys, , and , described by formulas ( 1)-( 3) with experimental values of corresponding fitted coefficients from Table 1.It can be seen that in the investigated temperature range the value of Δs increases by an order of magnitude!This increase is monotonous, but the change in slope is not, first increasing up to ≈1100 °C and then slightly decreasing.Such a dependence is due to the influence of several factors on the value of ∆s.As can be seen from formula (6), the temperature-increasing terms and α 2 (in the numerator) as well as the temperature-decreasing term (in the denominator) increase Δs, while the temperatureincreasing term (in the denominator) slows down the temperature increase of Δs.
Figure 2 shows that at low temperatures, up to about 600°C, the values for CMSX-4 and CMSX-10 are almost equal and they both increase with temperature.But with further temperature increase, the value of increases faster for CMSX-10 than that for CMSX-4, and at 1300 °C this difference reaches 54%.This difference is explained by the larger value of CLTR of CMSX-10 at high temperatures (see Fig. 1b), which is considered in Eq.( 6) as α 2 .Therefore, at high temperatures, a larger difference between the isothermal and adiabatic elastic characteristics should be expected for alloy CMSX-10.
In order to analyze the difference between isothermal and adiabatic elastic characteristics, an experimental temperature dependence for elastic constants, e.g., , is needed.Since the authors are not aware of such data for CMSX-10 published in the open literature, only alloy CMSX-4, which is widely used in gas turbine industry, will be considered below.In [6], the adiabatic elastic stiffnesses of CMSX-4 single crystals were measured in the temperature range from room temperature to 1300 °C using the resonance method.These data are given in the upper part of Table 2.The objective of the present work was to calculate the isothermal elastic characteristics of CMSX-4 and compare them with the corresponding adiabatic ones.
The calculations were performed in the following sequence: .First, using the data for and the well-known Eqs. ( 7)-( 9) relating and for cubic crystals, the adiabatic elastic compliances were calculated It should be noted that relationships (7)-( 9) are invariant with respect to the mutual permutation , that is, they can be used for the direct transformation , as well as for the inverse transformation (10) (11) (12) In the next step, the isothermal elastic compliances и were calculated by formulas (5), (6), and then using the obtained values of and the isothermal elastic stiffnesses and by inverse transformation (10), (11).As noted above, the adiabatic and isothermal compliances and are equal, and the same is true for the stiffnesses and , as follows from (12).Adiabatic and isothermal elastic moduli in the basic coordinate system with 001 axes were calculated from the values of the corresponding elastic compliances ( or ) using formulas where and ν are Young's modulus, shear modulus, bulk modulus of elasticity and Poisson's ratio, respectively.
All calculated values of elasticity characteristics are summarized in Table 2.
It follows from physical considerations that the values of adiabatic elastic constants characterize the elastic response of a body to its mechanical loading + thermal strain due to the change in temperature of the body during its elastic adiabatic expansion/contraction.Therefore, the relative difference between the adiabatic characteristics and the isothermal characteristics required in engineering calculations can be regarded as an error of determination of the required elasticity characteristics equal to: (17) where X S and X T are the adiabatic and isothermal characteristics under consideration, respectively.
Using relationships ( 10)-( 17), formulas for the relative errors of all commonly used elasticity characteristics were derived and summarized in Table 3, see appendix for details. 3 that all relative errors depend on the value , which is actually the error of adiabatic measurement of Young's modulus .Figure 3 shows graphs of temperature change of isothermal and adiabatic elasticity characteristics of CMSX-4, and their differences plotted using data from Table 2. Change of is shown in Fig. 3b.It can be seen that the dependence is non-monotonous.The value of is minimal at room temperature and is equal to 0.15%.With increasing temperature, increases to its maximum value of 0.97% at 1100°C and then decreases down to a value of 0.81% at 1300°C.Such a temperature change of is because its formula represents a ratio , where both the nominator ( ) and denominator ( ) increase with temperature, see Figs. 2 and 3a, respectively.At low temperatures, the factor prevails, while at high temperatures, the factor overcomes the factor due to significant elastic softening of the material.

It follows from the formulas in Table
From Fig. 3a follows that the isothermal and adiabatic elastic compliances do not differ much in the whole analyzed temperature range, and their difference in the graph is visually almost indistinguishable.The temperature dependences of the relative differences of the isothermal and adiabatic elastic compliances and are shown in Fig. 3b.It can be seen that is negative ( ), whereas is positive ( ).The shape of the curves and is similar to that of the curve , since and are directly related to as and , see Table 3.
The temperature dependences of the isothermal and adiabatic values of , and , and the relative differences , and are shown in Figs.3c and 3d.The isothermal and adiabatic shear moduli are equal because , that is, .It can be seen that the dependencies and are similar (with sign accuracy) to those of and , cf.Figs.3d and 3b.This is due to the simple relation of and to and , see ( 13), ( 16) and the following formulas for the relative differences, and , see Table 3.Since increases with temperature from ≈0.39 to ≈0.45 (see Fig. 3c), the ratio decreases from 3.6 to 3.2.The values of and are minimal at 20°C, respectively 0.15% and 0.54%, and increase to their maxima at 1100°C, respectively 0.98% and 3.4%, but then decrease to 0.81% and 2.6% at 1300°C.
Figures 3c, d also show the temperature dependences of the bulk modulus of elasticity and the relative difference .It can be seen that the difference between the values of and increases strongly with temperature (see Fig. 3c), their relative difference increases from ≈2% at 20°C to ≈28% at 1300°C.This large difference between and at high temperatures is explained by the form of formula (15) relating B to the elastic compliances and .In this formula, the sum in the denominator is acually the difference , since .For nickel and nicke-based alloys the value of is only slightly smaller , so a small decrease in and an increase in under adiabatic conditions significantly decreases , and consequently significantly increases the measured value of .This effect increases many times with increasing temperature since the ratio (double Poisson's ratio) increases, see Fig. 3c.As has been shown by experiment, e.g.[5,6], the Poisson's ratio of nickel-based alloys increases significantly with temperature, approaching 0.5 at temperatures close to solidus.The effect of the multiple increase of with temperature can also be explained by the form of the formula for in Table 3.Here, the ratio increases strongly with temperature because approaches 1, and the negative term enhances this effect.
The temperature dependences of adiabatic and isothermal elastic stiffnesses and relative differences and are shown in Fig. 3e, f.As noted above and are equal, that is, .From Figs. 3c, d, it can be seen that at room temperature, the difference between the values of adiabatic and isothermal elastic stiffnesses is not large: is larger than by 1.6% and larger than by 2.5%.However, with increasing temperature, this difference monotonically increases over the entire temperature range, up to 1300°C.At this temperature, ≈ 25% and ≈ 30%.At the maximum service temperature of the blade material of aircraft gas turbine engines (≈1100°C), and are equal to ≈16% and ≈23%, respectively.These large differences between and , and between and at high temperatures have the same reason as for и : the denominators of formulas ( 10) and ( 11) for and contain a factor similar to that in formula (15) for B. Due to this similarity of formulas (10), (11) with (15), the differences and are directly related to , see Table 3.
5. DISCUSSION As shown by the above analysis, the adiabatic and isothermal characteristics of nickel-based superalloys can differ significantly, and this is especially relevant to the high-temperature values of the elastic stiffnesses and , and the bulk modulus of elasticity B. Therefore, a reasonable question arises: what are the consequences of using adiabatic elastic constants instead of isothermal ones in engineering calculations?In order to clarify this question, let us analyze the relationship between the stress tensor and strain tensor .
For cubic crystals in the coordinate system 001 Hooke's law in matrix can be written as (18) From Eqs. ( 18) follows where , , and are, respectively, the volumetric strain, hydrostatic stress and deviatoric strain and stress determined as  In Eqs. ( 19)-( 21), only the bulk elastic modulus B depends on the measurement conditions (isothermal or adiabatic), whereas the shear modulus G and the difference do not: .Thus, the use of adiabatic elastic constants in engineering calculations affects the relationship between volumetric strain and hydrostatic stress σ.
Substituting adiabatic compliances instead of isothermal ones in (19) This effect of using adiabatic values of elastic constants is logical.When measured under adiabatic conditions, the values of elastic constants are influenced by isotropic thermal dilatation, which then appears in the description of volumetric strain in engineering calculations.In the case of inhomogeneous spatial distribution of hydrostatic stress , the use of adiabatic elastic constants instead of isothermal ones has an effect similar to introducing a field of thermal dilatation into the analyzed object.At low temperatures, this effect is small, but at high temperatures, typical for the service conditions of the blade material of aircraft gas turbine engines, it increases many times.
6. CONCLUSIONS 1.Based on the available experimental data for adiabatic elastic stiffnesses of single crystals of nickelbased superalloy CMSX-4 and the measured values of thermophysical characteristics of this alloy, isothermal elastic characteristics of CMSX-4 for a wide temperature range, from room temperature to 1300°C, have been calculated.It is shown that at high temperatures adiabatic values of such elastic characteristics as elastic stiffnesses , and bulk modulus of elasticity B are significantly larger than their isothermal values, by 16-23% at the service temperature of the blade material of aircraft gas turbine engines, ≈1100°C, and by 25-30% at the temperature of the hot isostatic pressing of blades, ≈1300°C.Such a large difference between the values of these isothermal and adiabatic characteristics is due to significant changes in the thermophysical properties of the alloy with temperature, as well as a high value of Poisson's ratio ν of nickel-based alloys, which approaches its limiting value of 0.5 with at high temperatures.
2. The use of adiabatic elastic constants instead of isothermal ones in engineering calculations affects the relationship between the volumetric strain and hydrostatic stress σ, and this effect is similar to introducing a field of thermal dilatation into the analyzed object.At low temperatures, this effect is small, but at high temperatures, typical for the service conditions of the blade material of aircraft turbine engines, it increases many times.
3. A comparative measurement of the thermophysical properties of alloys CMSX-4 and CMSX-10, and the subsequent calculation of the correction factor , which determines the difference between the adiabatic and isothermal characteristics of elasticity, showed a significantly larger value of for CMSX-10 at high temperatures.Thus, in order to calculate the values of isothermal elastic constants from the values of adiabatic constants one should use thermophysical characteristics (coefficient of linear thermal expansion, density and heat capacity) measured for the specific alloy in the relevant temperature range.

Fig. 3 .
Fig. 3. Temperature changes of isothermal and adiabatic elasticity characteristics (a, c, e) of CMSX-4 alloy, and their relative differences (b, d, f).In (a, c, e), the adiabatic characteristics are shown by square dots, the isothermal ones by round dots.
APPENDIX ACalculation of relative differences of adiabatic and isothermal characteristics of elasticity δX using formulas (10)-(17) 1.For elastic compliances: is, for small values of , the relative error increases strongly.3.For elastic stiffnesses: supported by the Russian Science Foundation under grant no.22-19-00126.OPEN ACCESSThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Table 2 .
[6]abatic (S) and isothermal (T) elastic stiffnesses, compliances and moduli of alloy CMSX-4.The values of are given in GPa and the values of s ij in TPa -1 .Adiabatic stiffnesses are taken from[6].

Table 3 .
Formulas for relative errors of elasticity characteristics of cubic crystals when measured under adiabatic