23

CONSTITUTIVE MATERIAL FORMULATIONS AND ADVANCED LIFE ASSESSMENT METHODS FOR SINGLE CRYSTAL GAS TURBINE BLADES

E.P. Busso1, M. Toulios2 and G. Cailletaud3

1Department of Mechanical Engineering Imperial College, London SW7 2BX, UK

2Department of Naval Architecture and Marine Engineering,

National Technical University of Athens, Athens 157 73, Greece

3Ecole Nationale Supérieure des Mines de Paris Centre des Matériaux, B.P.87, 91003 Evry Cedex, France

Abstract

Constitutive models for single crystal superalloys have been developed for the past fifteen years in response to the increasingly wider use of these materials to manufacture industrial and aerospace gas turbine blading. A number of model formulations are now available that aim to predict correctly the deformation behaviour at loading conditions similar to those experienced in service. Some of the most relevant work in this field is reviewed in this paper. In engineering practice, the complexity of the material models used depends on the required level of accuracy and the ability to effectively perform computationally intensive FE analyses, due to the complex geometry of a cooled blade. However, it is argued that a constitutive formulation should take into account various features of the microstructure to predict correctly the effect of heterogeneities in the material that can be critical to the life of a component. Certain key aspects in the design assessment of cooled blades are also discussed, in particular the implications of the choice of constitutive models in the blade design procedure. Keywords: constitutive models, multi-scale modelling, single crystal superalloys, cooled blade design

Introduction Land-based and aerospace gas turbines often rely on more than one row of single crystal blades to expand and speed-up the hot gases from the combustor while they turn the rotor. Nickel based superalloys have become widely favoured as the material of choice for turbine blades since they offer good high temperature properties. The introduction of a new superalloy in an existing or future blade design is a complex and time-consuming process. The comprehensive characterisation of the mechanical behaviour of the alloy contributes significantly to the duration and cost of this process. Often the development cycle time is shortened by using high safety factors, which lead to a more frequent replacement programme to address the uncertainty and/or limitations in the life estimation methods. However, this conservative, experience-based, approach of the past is gradually being substituted with a much shorter and accurate computer-based modelling approach [1][2]. With each new generation of blade materials, advanced materials models are required to predict accurately their high temperature deformation behaviour, including the effects of changes in phase constitution as a function of chemical composition, temperature and time. The need for advanced constitutive models is highlighted by the fact that a major contributing factor to the acceleration of creep rates and to fatigue failures in superalloy components is the localization of the inelastic deformation caused by the heterogeneity of the material microstructure. Most advanced constitutive models developed to date for superalloys aim at predicting such

Proc. 7th Liege Conf. On Materials for Advanced Power Engineering, Part I, Sept. 30, 2002, Belgium, J.Lecomte-Beckers et al. (eds.), Forschungszentrum Julich, GmbH (publ.), (2002), pp. 23-42.

24

complex behaviour to assure the accuracy of hot section component design and life assessments. This paper initially reviews recent developments in the formulation of constitutive models to describe the stress-strain behaviour of single crystal (SC) superalloys. The review is divided in three parts. The first one addresses several formulations developed during the past 10-15 years which, although physically based, do not explicitly take into account the heterogeneities that arise in SC superalloys. In the second part, more recent constitutive approaches, which incorporate the evolution of the microstructure at different length scales to deal with the effects of a heterogeneous microstructure, are briefly described. Finally, the paper describes certain key aspects in the design assessment of cooled blades and discusses the implication of the choice of constitutive models in blade design. Microstructural Considerations and Constitutive Frameworks In Ni-base superalloys, material heterogeneities exist at both the microscopic and mesoscopic levels. At the microscale, they can occur due to the changing morphology of the ~ 0.5-1 µm γ' precipitates during service, i.e. the so-called rafting process, and to the breakdown of uniform patterns of plastic deformation into localised deformation modes (i.e. shear bands) from which cracks can initiate, e.g. see [3]. At the mesoscale, heterogeneities may exist in the material in the form of either casting and processing defects (e.g., ~10-30 µm diameter casting micropores , inclusions), or as 20-200 µm eutectic regions, which generally precipitate during the final stages of solidification (e.g. see [4]), occupy interdendritic regions and are composed of a high volume fraction (>90%) of γ' precipitates of irregular shape. During service, local conditions around these homogeneities such as stress-state, anisotropy, temperature and interaction with a free-surface can give rise to microcracks initiating, e.g. from the circumference of embedded casting defects, following an incubation or growth period. Coalescence of these microcracks often leads to failure at the macroscopic level [5][6]. Constitutive models developed to predict the high temperature behaviour of single crystal superalloys follow either a Hill-type or a crystallographic approach. As a common feature, they treat the material as a continuum in order to describe properly visco-plastic effects in the temperature range of interest. Hill-type approaches (e.g. [7][8][9]) are based on a generalisation of the Mises yield criterion proposed by Hill [10] to account for the non-smooth flow potential surface required to describe the anisotropic flow stress behaviour of single crystal materials. In visco-plastic constitutive formulations based on crystallographic slip, the macroscopic stress state is resolved onto each slip system following Schmid's law. Depending on the relative orientation of the slip system with respect to the macroscopic principal stress directions and to the magnitude of the resolved shear stress, the slip system may be activated and slip produced. Internal state variables are generally introduced in both formulations to represent the evolution of the microstructural state during the deformation process. Although recent developments in these two approaches have now reached an advanced stage, the major improvements have been made by crystallographic models due to their ability to incorporate complex micro-mechanisms of slip within the flow and evolutionary equations of the single crystal models. These include the effects of dislocation interactions [11][12], strain gradient phenomena [13][14][15], precipitate morphologies [13][16] and their spatial

25

arrangements [17], and general anisotropic visco-plastic behaviour (e.g. [7][11][12][18][19] [20][21][22][23]). General form of the flow and evolutionary equations A rate-type formulation for the time rate of change of the stress tensor, T, under small strains and rotations and isothermal conditions gives,

[ ] - iEET DDD = (1)

where ED and iED are the total and inelastic strain tensors, respectively, and L is the fourth order elastic anisotropic moduli. In an anisotropic constitutive framework, the macroscopic deformation evolves from a flow rule that is derived from a flow potential function, Ω . The evolution of the inelastic anisotropy of the material, represented by iED in Eq. 1, is obtained from

T

E∂Ω∂= (2)

The particular form of the flow potential function varies between the constitutive approaches, as it will be discussed next. Crystallographic approach In a crystallographic formulation, the time rate of change of the inelastic strain tensor is derived from the kinematics of dislocation motion. It can be shown that,

∑=α

ααγm

i PE , with ( )ααααα nmmnP ⊗+⊗=21 (3)

Here, αm and αn are unit vectors defining the slip direction and the slip plane normal for the slip system α, respectively, αγD , represents the rate of slip in the slip system α, the summation extends over the αm active slip systems and the symbol ⊗ denotes the outer product of the vectors (i.e. the components of the orientation tensor αP are, ( )ααααα

ijjiij mnmnP += 5.0 ). For the case of superalloy single crystals, inelastic deformation is generally assumed to take place by crystallographic slip along two families of slip systems, namely twelve octahedral 111<110> and eight cubic 100<100> systems. (Note that, even though the latter system has not experimentally identified from TEM investigations, its inclusion is necessary to predict consistently the effect of crystallographic orientation on the macroscopic response of the superalloy, e.g. see [11][12][20][23]). The slip rate in Eq. 3 can, in its most general form, be functionally expressed as,

αααααα θτγγSnSSS ,......,,,,ˆ 21

DD = (4)

where θ is the absolute temperature, ατ is the resolved shear stress in the slip system defined by,

T : Pαατ = (5)

and αiS , for i=1, nS, represents a set of internal state variables for the slip system α. The

latter accounts for the current microstructural state of the dislocation or obstacle network on

26

each slip system α. Their evolutionary behaviour should be linked to the dominant hardening and recovery processes in the single crystal (e.g. [21]). For crystallographic formulations which do not incorporate deformation gradient - dependent effects, the time rate of change of each internal slip system variable, α

iSD , can, in its most general form, be expressed as,

αααααα

αααααα

αααααα

θγ

θγ

θγ

SSS

S

S

nnn

ni

n

SSSSS

SSSSS

SSSSS

,......,,,,ˆ

,......,,,,ˆ

,......,,,,ˆ

21

2122

2111

=

=

=

(6)

The set of non-linear differential equations 1, and 3 to 6 constitute the complete crystallographic formulation, which must be solved numerically. This is typically done using implicit Newton-type algorithms (e.g. see [15]). Hill-Type approach In a Hill-type anisotropic framework, the macroscopic deformation evolves from a flow rule that is derived from a Hill-type flow potential function, ψ [10]. The time rate of change of the stress tensor, T, is given by an analogous relation to that of Eq. 1. The evolution of the inelastic anisotropy of the material, iED in Eq. 1, is obtained, in component form, from

'~

ij

i

ij

iij TT ∂

∂=∂

Ω∂= ψεε DD (7)

where 'ijT represents the deviatoric component of the stress tensor, T, and iε~ the magnitude of

the equivalent inelastic strain rate. In their most general form, ψ and iε~ can be functionally expressed in terms of T, the temperature θ, and the current microstructural state represented by a set of one macroscopic tensorial, B , and nS macroscopic scalar internal state variables,

SnSSS ,,, 21 , respectively. (Note that the bar over the symbols indicates the macroscopic nature of the internal state variables). Then,

SnSSS ,,,,,ˆ 21 ,BT θψψ = (8)

Sn

ii SSS ,,,,,~~21

,BT θεε = (9)

It should be noted that the internal state variables in Eqs. 8 and 9 can be considered to be the macroscopic equivalent of those introduced at the slip system level in a crystallographic framework, viz. Eq. 4. The formulation is completed with the evolutionary equations for each macroscopic internal state variable. In their most general form, one can write,

27

SSS

S

S

S

ni

nn

ni

ni

ni

SSSSS

SSSSS

SSSSS

SSS

,......,,,,,~ˆ

,......,,,,,~ˆ

,......,,,,,~ˆ

,......,,,,,~ˆ

21

2122

2111

21

B

B

B

BBB

θε

θε

θε

θε

=

=

=

=

(10)

As in the crystallographic approach, the set of non-linear differential equations given by Eqs. 1 and 7 to 10 must be integrated numerically using an appropriate integration scheme (e.g. see [24]). In the next section, some examples of crystallographic and macroscopic constitutive formulations developed for single crystal superalloys are given. Constitutive Models for Superalloy Single Crystals Crystallographic Models The amount of modelling work done on superalloys has been extensive in the last 15 years or so. In this section, only an outline of some of the most relevant work in this area will be given. Most superalloy models based on an internal slip system variable framework, such as those of [11][12][14][20][21], describe the microstructure evolution through two internal state variables per slip system: (i) a macroscopically average slip resistance Sα (sometimes confusingly referred to as an “isotropic hardening variable”, due to its non-directional nature), and (ii) a back or internal stress Bα, which represents the current polarisation of the dislocation/obstacle network associated with the macroscopically observed kinematic hardening. Thus, Eq. 4 becomes

ααααα θτγγ BS ,,,DD = (11)

An alternative and more physically intuitive way of interpreting the inherent differences between flow rules is found by inverting Eq. 11 to obtain the dependence of the flow stress on temperature, strain rate and microstructural state [25]. Thus, one can obtain a general expression of the form,

0 ,,,ˆ cBcScBSF BSV +++= ααααα θγτ (12)

Here, 0c is a threshold value, Sc and Bc are scaling coefficients of order unity, and VF is a generic function. Until relatively recently, almost all crystallographic formulations relied on power law functions of the resolved shear stress for Eq. 11, with and without a threshold stress. A general power law form widely used (e.g. [20]),

28

( )ααα

εαθα τ

τγγ Bsign

S

cBe

n

kG

−−−

=

− 0

0 0

DD (13)

where temperature effects are introduced through the Arrhenius term, and the activation free energy, 0G , the constants 00 ,cγC and n, and the initial value of αS (note that the initial value of 0| 0 ==tBα ) constitute the minimum number of material constants per slip system in the simplest and most commonly used form of Eq. 11. It can be shown that the inversion of Eq. 13 gives

0

/1

0

0

cBeSn

kG

++

=

αθα

αα

γγτD

D (14)

Even though Eq. 14 is generally considered to be “a classical one”, it also highlights the two main problems with this type of flow rule. Firstly, the viscous term, i.e. first term in Eq. 14, is also affected by the current state through the value of αS . Such coupling poses problems when calibrating the rate dependent response of the constitutive model. Furthermore, some rate-dependent activated mechanisms, such as lattice friction, are known to be independent of the current microstructural state, thus in such cases θγ α ,ˆ VV FF = only. Secondly, the presence of an athermal and rate independent threshold value, viz. 0c in Eq. 14, which is independent of the current microstructural state, is inconsistent with most strengthening mechanisms. In the work of Meric et al. [11][12], a different power law relation is used for the slip rate dependence on the resolved shear stress in each slip system, in that the slip resistance is written in the numerator, in contrast to Eq. 12. Then

( )αααεα

θα ττ

τγγ Bsign

SBn

−−−

=

−

0

kG

0 ˆ e

0

DD . (15)

By inverting Eq. 15 and expressing it in the form of Eq. 12, one finds

ααθα

α

γγττ BSe

n

kG

++

=

/1

00 ˆ

0

(16)

Equation 16 provides a more physically meaningful interpretation of the strengthening mechanisms contributing to the overall flow stress of the single crystal than Eq. 14: the first term describes uniquely the viscous effects, and the second and third terms the contributions from the current microstructural state. Equation 15, in conjunction with the corresponding evolutionary equations, was used by Meric et al. [11][12] to describe the SC behaviour of the nickel base superalloy AM1. The work focused on predictions of uniaxial monotonic and cyclic isothermal tests at 950 oC. Figures 1(a) and (b), taken from [26], show a comparison between experimental and predicted cyclic responses for the superalloy single crystal AM1 at 950oC. It can be seen that correct predictions are obtained for [111] and [101] cycles (a) with and (b) without peak tension strain holds. The Meric et al.’s [11][12] constitutive single

29

crystal formulation was also numerically implemented into a finite element code (e.g. see [27]) so that the experimentally measured deformation of the cross sectional profiles of tubular specimens subjected to torsion could subsequently be compared with numerical FE predictions. For highly symmetric orientations, viz. <001> and <111>, good agreement was found between experiment and predictions.

(a) (b)

Fig. 1. Comparison between experimental (symbols) and predicted (lines) cyclic responses for the superalloy AM1 at 950oC: (a) with and (b) without peak tension strain holds along [111] and [101] orientations, respectively [26]. More recently, Golan et al. [28] explored the possibility of using a modified version of the Norton creep power law with an internal stress or back stress internal variable to predict the uniaxial creep behaviour of CMSX-2 at 982oC, along crystallographic orientations close to the [001] axis. It was found that an empirical relation exists between the parameter multiplying the power law term and the exponent n. The existence of such a relation indicates that the activation energy for the creep process is continuously changing as deformation of the single crystal progresses, reaching a peak when a rafted microstructure is obtained. However, the empirical and one-dimensional nature of this type of model limits its usefulness. One important limitation of the widely used power law relations, such as those in Eqs. 13 and 15, is that at a given temperature and at a constant internal state, the exponent n imposes a constant strain rate sensitivity in the material behaviour, irrespective of the stress level. Thus, it is a limitation when describing the behaviour of materials that exhibit non-linear strain rate sensitivities over stress ranges typical of service. Nevertheless, power law based flow rules have been shown to be suitable under well-defined stress ranges. The effect of this constraint can to some extent be reduced by introducing, at the expense of additional material parameters, a static recovery term in the evolutionary behaviour of the back stress (e.g. see [11][12][20]). For instance, in the work of Jordan & Walker [20], uniaxial tests on PWA 1480 at 889oC along the symmetric orientations <001> and <111> were reasonably correlated using a flow rule of the form given in Eq. 13. Torsion tests, designed to induce a biaxial stress state,

30

provided reasonable correlation only if both octahedral and cubic slip systems were assumed active. One way of introducing non-linear strain rate sensitivity into the crystallographic formulation without allowing the power law exponent to be a variable or without rather artificially introducing a static recovery term, is by using either hyperbolic (e.g. [29]) or exponential functions of the resolved shear stress in the flow rule. The former is a purely phenomenological way of introducing this effect, whereas the latter is more useful as it allows for the dependency of the activation free energy on stress to be readily incorporated [21][25]. More recently, the crystallographic framework initially proposed by Busso [25] (see also [14][15][21][23]) for NiAl single crystals was further developed to describe the behaviour of superalloy single crystals. Here, the flow rule relies on a stress-dependent activation energy expressed in terms of a macroscopically average slip resistance Sα, and a back or internal stress Bα. The flow rule is expressed in terms of an exponential function as,

( )ααααα

α τµµτ

µµτθ

γγ BSB

kF

qp

−

−−

−−= sgnˆ

1exp00

000 (17)

where µ, µ0 are the shear moduli at θ and 0 ºK, respectively, and F0, 0τ , p, q and 0γD are material parameters. Likewise, an inversion of Eq. 17 allows the flow stress to be expressed as a function of its contributing terms. Here,

ααα

µµ

θθ

µµττ BS

pq

++

−=

0

/1/1

000 1 ˆ (18)

where

( )αγγθ

/Ln1

kF

0

00 = (19)

It can be seen that Eqs. 18-19 provide a similar physically intuitive strengthening framework as Eq. 16 since the first term describes uniquely the strain rate and temperature effects, and the second and third terms the contributions from the current dislocation resistance and internal stresses. One shortcoming of Eq. 17 is that the slip rate for 0=ατ is not zero but equal to a very small residual value (typically < 10-15 1/s.). Nevertheless, for most practical purposes, this residual value has no effect on the overall predicted deformation of superalloy components and it can, in fact, easily be forced to be zero when integrating Eq. 17 numerically. In contrast to power law relations, the exponential form of Eq. 17 enables the non-linear stress rate sensitivity exhibited by superalloy single crystals to be described correctly. This point is illustrated in Fig. 2, which shows the strain rate versus the steady state flow stress curves obtained for a <111> oriented CMSX4 specimen at 950oC and the corresponding data [23]. In general, the strain rate sensitivity (which for a given state, it is controlled to a great extent by the power law coefficient n) predicted by power law relations of the type given in Eqs. 13 and 15 would underestimate the deformation at both very high and very low stress levels. This

31

problem can be overcome by using an exponential flow rule of the type shown in Fig. 2. These results also highlight the critical nature of the model parameter calibration, in particular the stress range at which this is done. It is also worth noting that the stress levels seen in turbine blades can be very low, thus the material model calibration must rely on test data obtained under such representative stress conditions. The evolutionary behaviour of the slip system variables is typically defined within a hardening-dynamic recovery format. The form and type of equations vary within formulations and it would be outside the scope of this work to present the details. As a typical illustration, those given by [14][15][21][25] will be shown. Here, the evolutionary behaviour of the overall slip resistance is defined as,

[ ]∑=

−−=α

β

αββαβα γδm

DSS SSdhS1

0 || )( DD (20)

where, β0S is the initial value of βS , dD is a dynamic recovery parameter, and αm the total

number of slip systems. In Eq. 20, αβδ S is the latent hardening or interaction function [20],

αβαβ δωωδ )1( 21 −+=S (21)

where αβδ is the Kroneker delta so that 021 ==ωω corresponds to self-hardening, and 121 ==ωω to Taylor hardening.

100 1000 1000010

-13

10-11

10-9

10-7

10-5

10-3

10-1

. ~ Creep data

Monotonic dataData

950 oC

900 oC

850 oC

Predictions

ε <11

1> [s

-1]

~ σ<111> [MPa]

Fig. 2. Evolution of strain rate vs. steady state stress for a <111> oriented CMSX4 specimen at 950oC: non-linear strain rate sensitivity predicted by the exponential slip rate relation Eq. 17 [23].

32

The back stress evolves according to the well-known Armstrong-Frederic hardening-dynamic recovery framework,

r

SDB BrBrhB ααααα γγ −−= DDD (22)

where Bh is the hardening coefficient, αSrr DD ˆ= a dynamic recovery function expressed in terms of the current overall deformation resistance, αS [21][25], and Sr and r, static recovery parameters. The model was calibrated to predict the visco-plastic behaviour of single crystal superalloy CMSX-4 using a database of monotonic, cyclic and creep data. No static recovery was considered for simplicity. It was then validated by comparing the predicted response with experimental data obtained from complex thermal-mechanical histories and multi-axial stress states. Figure 3 shows typical predictions of monotonic and cyclic responses obtained experimentally from CMSX4 at 950 and 850oC, respectively [30].

0 2 4 6 8 100

500

1000

1500

...

SC Model Data ε = 10-21/s ε = 10-31/s ε = 10-41/s

~ σ

<00

1> [M

Pa]

~ ε<001> [%]

(a)

0.8 1.0 1.2 1.4 1.6 1.8

-800

-400

0

400

800ε = 10-31/s, 850 C

Data

~

σ <11

1> [M

Pa]

~ ε<001> [%]

(b)

.

SC Model

o 950 C o

.

Fig. 3. Comparison between measured and predicted (a) <001> monotonic response at different strain rates and 950oC, and (b) <111> steady state cyclic response at 850oC [30] Continuum damage mechanics (CDM) concepts have also been used to incorporate the effects of microstructure evolution on the tertiary creep behaviour of superalloys (e.g. through a strain softening mechanism linked to the accumulation of dislocations [18][22][29][31]). CDM – based crystallographic models contain one or more scalar damage variables at the level of the slip system, in addition to or instead of the directional and non-directional hardening variables given in Eq. 11. These models are generally calibrated from uniaxial creep data obtained at different temperatures and stresses, hence their predictive capabilities

33

are limited to creep deformation and monotonic stress-strain responses at low strain rates close to the minimum ones obtained in creep testing [29]. The prediction of transient monotonic and cyclic stress-strain conditions requires an accurate description of the Bauschinger effect, which can be only clearly identified from cyclic data. A further limitation is the difficulty in formulating a three-dimensional version of CDM models and the mesh sensitivity of FE results in regions where deformation may localise. A phenomenological approach that has received some attention is based on the use of appropriate mathematical representations of the <001> and <111> creep curves. These models are therefore primarily calibrated from uniaxial creep data and their parameters are related, in a relatively simple manner, to various features of the creep curve. Multiaxial formulations are obtained via Eq. 3 using distinct <001> and <111> parameters. In [32], the creep curves are fitted with a Graham-Walles type equation. This uses a summation of stress and time power law terms, although in [32] an additional term is used and the temperature dependence is now introduced via an exponential form. One of the main differences between the Graham-Walles equation and the CDM approach of [29], is that the latter combines primary and tertiary creep in product form, which implies that primary and tertiary creep mechanisms act in parallel. The creep model proposed in [33] also incorporates a back stress state variable which evolves according to a hardening-static recovery law (with no dynamic recovery and r =1 for the static recovery term in Eq. 22). The creep rate expression also includes a tertiary creep softening term that is also combined in product form. The use of these models to predict the transient monotonic and cyclic stress-strain response can also lead to inaccuracies. However, it appears that there is the potential to use this approach for creep-dominated behaviour, see [33]. Although it is well known that directional coarsening of γ' precipitates strongly affects the creep and fatigue behaviour of the material, e.g. [34] [36], there has been only limited amount of work addressing this issue within a crystallographic framework. This is due mostly to the fact that raftening is associated with a loss of cubic symmetry. To accommodate such changes within a slip system based framework requires that groups of slip system within the same family (e.g. octahedral) be calibrated differently. As it will be discussed below, the incorporation of raftening effects may be done more easily using a macroscopic single crystal framework. Hill-Type Models Several macroscopic phenomenological formulations have been developed for single crystal superalloys (e.g. [7][8][37][38]). Most of them are modifications of isotropic macroscopic formulations, with a different criterion to account for the cubic symmetry of the material. The approach of Nouailhas and Freed [8] relies on fourth order tensors to define both the elastic and inelastic anisotropy of a superalloy single crystal. Despite the simplicity of the constitutive equations and the reduced number of state variables, which offers some advantages for finite element structural calculations over crystallographic based models, such type of models have been found to have limited predictive capabilities for some particular cases such as under torsional loading. As they rely on a Hill type criterion, only one free parameter can be specified for the shear behaviour relative to the tensile behaviour. Thus, it was found in [39] that, for the case of a tubular specimen subjected to pure torsion, a uniform stress-strain distribution is predicted along the circumference of the specimen. This is

34

inconsistent with both Schmid’s law and the experimental evidence reported in [40]. In follow-up work, Nouailhas et al. [38] proposed a new improved version of the model presented in [8], whereby the potential function used to describe the initial material anisotropy was now expressed in terms of nine stress invariants. This new model showed improved predictive capabilities and overcame the limitations exhibited by the previous model as cyclic data and slip traces obtained from CMSX-2 torsional tests at room temperature and at 950oC were reasonably well predicted by the model. Schubert et al [9] relied on an orthotropic Hill-type potential, with the anisotropic coefficients linked to the evolving morphology of the γ’ precipitates, to incorporate the effects of precipitate raftening within the macroscopic behaviour of the superalloy CMSX4. The model was calibrated from <001> and <111> creep data and microstructural observations in order to define the stress-temperature regime where raftening occurs. In the absence of raftening, the orthotropic potential used reduces to the cubic form of Hill’s potential. The model was found to correctly predict the orientation dependence of the minimum creep rate at 950°C, where raftening readily takes place, and it has successfully simulated the <100> and <111> creep deformation behaviour. Busso & McClintock [7] studied the anisotropic creep behaviour of the superalloy single crystal CMSX4 using a Hill-type anisotropic creep (or flow) potential for a cubic crystal and the associated flow rule. The anisotropic parameter in the Hill flow potential was calibrated from uniaxial creep data along <110> and <100> crystallographic orientations at different temperatures and stress levels. No hardening or softening behaviour was incorporated in the model. Even though the model was not calibrated against complex multiaxial data, a good representation of the anisotropy of the steady state creep behaviour of the superalloy was obtained. Advanced Constitutive Models for Superalloy Single Crystals The main driving force behind the current and future research on superalloy characterisation and modelling is the development of constitutive formulations that can incorporate features of the micro-mechanisms of deformation and damage. This includes the effects of microstructural evolution during service occurring simultaneously at different scales in the microstructure and coupling with kinetics processes such as oxidation. Such approaches are referred to as multi-scale and multi-physics, respectively. For the case of superalloys, most of the current models are unable to predict the effect of local variations in the precipitate volume fraction on the local material behaviour. Recently, precipitate volume fraction effects were quantified for a range of temperature and strain rate conditions using a strain-gradient crystallographic framework and periodic unit cell-based FE analyses [14][15]. These results have been incorporated into a state variable crystallographic formulation to account for experimentally observed precipitate volume fraction and size effects in a single crystal nickel-base superalloy [23].

35

o 3 1

oθ = 850 Cε = 10 s

001

ε [%]001

001

V = 0.68f

V = 0.58f

Data

[MP

a]

Fig. 4. Effect of precipitate volume fraction on the <001> monotonic response of CMSX4 at 850oC and 10-3 1/s. Symbols represent experimental data for the 68% vol. fraction case [23].

The resulting crystallographic formulation incorporates an explicit link between the γ’ precipitate population at the microscale, and the behaviour of the homogeneous equivalent material at the macroscale. This link is introduced through the dynamic recovery function dD and the initial microstructural state, β

0S , in Eq. 20. Thus they depend on the characteristics of the current precipitate population as follows,

fDD Vlldd ,/,ˆ0θ= (23)

fVllSS ,/,ˆ000 θαα = (24)

where 0/ ll is the precipitate size, l, normalised by a reference mean value, 0l , and fV the precipitate volume fraction. Equations 23 and 24 were calibrated from FE analyses of periodic unit cells at the microscale containing the individual precipitates [14][15]. Typical predictions of the monotonic uniaxial behaviour of CMSX4 at 8500C and 10-3 1/s are shown in Fig. 4 for two different γ’ volume fractions, namely 58 and 68 %, together with experimental data for the latter case. These results show that a 10 % reduction in volume fraction results in a 40 % decrease in the superalloy steady state flow stress at this temperature and strain rate. Work to be presented in this conference [41] will describe recent progress made in developing a multi-scale crystallographic framework to characterize the effect of different γ-γ’ eutectic regions on the mechanical behaviour of nickel-base superalloys. As the volume fraction of eutectic regions in the superalloy can be controlled by heat treatment, this type of work can have important implications in the selection of adequate homogenisation heat treatments so as to optimise the mechanical properties of the single crystal superalloy.

36

On the Design Assessment of Industrial Gas Turbine Single Crystal Blading Thermo-mechanical response of cooled blades The service cycle of an industrial gas turbine (IGT) consists of rapid transient periods, during start-up and shutdown, and long intervals of steady-state operation. The duration of the transients varies according to the power output of the engine. A small IGT (e.g. 6-12MW) can reach operating speeds of 12,000 rpm within 20 to 80 seconds. A large IGT (e.g. 200MW) will typically take between two to three minutes to reach a speed of 3,000 rpm. There are also varying requirements concerning the steady state operation of an engine. Accordingly, a large GT plant that contributes to base load operation can be subjected to a daily shift. A smaller GT that provides power to a small industrial unit may operate at steady state on a weekly basis. Cooled high-pressure turbine blades are subjected to complex thermo-mechanical loading. The mechanical loads consist of centrifugal and gas forces, with the latter being the result of pressure differences across the aerofoil surface and between the internal cooling air and the external hot gases. The thermal stresses are induced by non-uniform temperatures in the aerofoil due to temperature differences between the cooling air and hot gases. This difference varies during the transients and peaks during start-up. In view of the complexity of the loading and, in particular, the geometry of a cooled blade, the stress distribution in the aerofoil can only be estimated through detailed inelastic FE analysis. It is important to highlight two key aspects of blade design. The centrifugal loads introduce high radial stresses that are tensile in nature, while the thermal loading induces biaxial compressive stresses on the hot external surface of the aerofoil. It is also instructive to distinguish further between the response of small and large blades when addressing the key issues related to blade design procedures [42]. For instance, in a large blade with a typical 3 mm wall thickness, the thermal stresses are sufficiently high so that the biaxial stress state on the surface of the blade is compressive. In contrast, in a small blade with a 1mm wall thickness, the thermal stresses are lower and the radial stresses are predominantly tensile while the tangential stresses are compressive. It should be noted that these remarks do not account for stress redistributions during steady state operation. The aerofoil section of a cooled blade has a number of geometric features that can lead to a significant increase in the local stresses. Such features include the fillet regions joining the aerofoil to the platform or the shroud, cooling passages with ribs, and infringement cooling holes. Due to computational constraints, when constructing a three dimensional FE model of the blade to determine its (overall) inelastic stress-strain response, such features are either left out or not modelled with the required detail to capture the peak (localised) stresses. In such cases, elastic FE analyses of candidate blade components have shown that the stress state in the aerofoil is well below yield and that during each load cycle the aerofoil is only experiencing elastic-creep deformation. This is shown in Fig. 5 in terms of an upper bound curve of the resolved shear stresses in typical IGT blades with metal temperatures between 550 to 950ºC. The curve has been constructed by plotting the maximum octahedral and cubic shear stresses in the aerofoil at each temperature. The upper bound values are compared to the critical resolved shear stresses (CRSS) for CMSX-4 which have been obtained from <001> and <111> tensile data at 0.6%/min (dashed/dotted lines) and 6%/min (dashed lines) [43].

37

500

0

100

200

300

400

600 700 800 900 1000

Temperature C

ο

Stre

ss, M

Pa

Upper bound curve

CRSS - Octahedral

CRSS - CubicSt

ress

, MPa

Angle φ

Fig.5: Upper bound curve of the elastically calculated resolved stresses in typical IGT aerofoils compared to the CRSS values of CMSX-4. The inset, based on the simplified method of [42], shows the graphs of the octahedral and cubic stresses that give the maximum values at a cross-section in the aerofoil root of a large blade. The angle φ measures the in-plane rotation from the secondary [100] crystal axis. The peak stresses present in the stress concentration regions mentioned above can lead to creep-fatigue problems, particularly in applications that are expected to experience a large number of start/stop cycles. For example, the elastic stress concentration factor at an infringement cooling hole can be greater than two, depending on the combination of the far field radial and tangential stresses. Also the peak stress, tangential to the hole surface, can be tensile or compressive. That is, in a large blade the far field compressive stresses lead to a compressive stress while, in a small blade, it is mostly tensile but changes to compressive in the upper half of the aerofoil. Such detailed estimates are currently possible using submodeling FE techniques which, in practice, are only used in elastic analyses. Otherwise it is necessary to specify not only the magnitude and distribution of the transmission loads from the adjoining structure to the submodel, but also their evolution during cycling. In the inelastic regime, it is also more difficult to identify a representative submodel that has a negligible effect on the overall solution, particularly during steady state operation where creep deformation can lead to widespread redistribution of the stresses. Current and future applications of inelastic analysis methods The above discussion points to a three-level application of inelastic analysis methods in the life assessment of turbine blades. The first level, believed to be current practice in blade design, involves the computation of the stresses and strains in the aerofoil section of the blade during steady state (i.e. constant load) operation. Since the geometrical features that lead to high stress concentration factors cannot be efficiently included in such analyses, an FE model of the aerofoil is experiencing pure creep deformation, as shown in Fig. 5. Thus, it is the creep behaviour of SX superalloys which should be reliably predicted. This conclusion has

38

prompted the development of some of the creep calibrated models mentioned above, e.g. [9][33], and also the acquisition of medium to long-term deformation data that are closer to IGT operating conditions, e.g. [43]. It should also be mentioned here that peak stresses in the aerofoil will generally redistribute and relax fairly quickly. Thus the average stresses in the blade should be such that rupture will occur in excess of 105 hours. Accordingly, the emphasis here at this first design level is to obtain long-term engineering predictions neglecting various microstructural features that can also be critical to the life of the component. The second level of application concerns the inelastic analysis of a blade that includes sufficient refinement, either in the geometric or the FE idealisation, to capture the local peak stresses in some of the stress concentration features mentioned above. In this case, the behaviour is no longer elastic-creep since additional visco-plastic deformation occurs in the initial loading (i.e. the first start-up) and possibly also during the ensuing loading cycles (in addition to that incurred during each dwell period, i.e. in steady state operation). It is possible here to provide an indication of the allowable strain levels during load cycling, assuming a minimum design life of 3000 cycles. On the basis of CMSX-4 data from continuous LCF tests [43], this implies that the total strain range at typical stress concentration regions should be less than 1.2% to 1.4%. For these strain values, as Fig. 6 shows, the amount of visco-plastic strain incurred during cycling is small. Note that using failure data from LCF tests that include hold periods will reduce the allowable strain ranges even further. The emphasis therefore firstly shifts to predicting accurately the visco-plastic deformation incurred during the initial monotonic loading as in Fig. 3(a). Secondly, during load cycling, the constitutive models should be capable of correctly representing the mean stress relaxation at low strain values. Computational constraints still limit this application in engineering design practice. However, recent advances in numerical techniques and parallel processing are expected to lead to wider usage, e.g. [44].

0

300

600

900

1200

1500

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

Modified Total Strain Range, %

Stre

ss R

ange

, MPa

<001> data<111> data<011> datamean fit: all datamean fit: <001> datamean fit: <111> data

Fig. 6. Cyclic stress-strain curve constructed from CMSX-4 half-life LCF data at 950°C and 6%/min [43]. The modified strain range, used to eliminate the effect of anisotropy (up to around 1.4%), equals the applied strain range multiplied by the ratio of the Young’s modulus of the specimen over the Young’s modulus in the <001> orientation.

39

The third level of application involves the use of advanced single crystal models, such as those outlined above, which are capable of properly accounting for the effects of the initial and the evolving microstructure on the deformation and damage behaviour. Clearly, the hot metal temperatures of gas turbine blading, coupled with the complexity of the resulting stress state, can lead to drastically different changes in the microstructure. For example, the compressive stress state in a large blade will induce different coarsening of the γ΄ precipitates than that found in typical creep tests in tension. A different precipitate morphology will result in significantly different deformation rates, e.g. [45]. Further differences appear to arise at the mesoscale, e.g. specimens loaded in compression show little or no evidence of micro-cracking, originating at either micropores, carbides or in eutectics, as observed under a tensile load [45]. Advanced single crystal models, e.g. of the type outlined above, are now beginning to be used in the analysis of actual blade components operating under steady-state conditions [46]. Concluding Remarks This paper has reviewed a number of constitutive material formulations for single crystal superalloys that have been developed during the past fifteen years or so. Both crystallographic and macroscopic (Hill-type) approaches have been considered, although the former was discussed in greater detail in view of its wider acceptance. Emphasis was given to constitutive formulations which rely on internal state variables to describe the microstructural evolution during service. The need for advanced constitutive models was highlighted in view of the (intrinsic) microstructural and deformation-induced heterogeneities which are present in the material and which can have a significant effect on component lifetimes. A three level application of the constitutive models in the design of cooled blades has been put forward, based on certain aspects of their thermo-mechanical response and the ability to carry out computationally intensive FE analyses. References [1] Seth, B.B., Superalloys – The Utility Gas Turbine Perspective, in Superalloys 2000,

T.M.Pollock et al. (eds), The Minerals, Metals & Materials Society (TMS), USA (publ.), 2000, 3-16.

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[46] Busso, E.P., O'Dowd, N.P, and Dennis, R., A Crystallographic Formulation for Superalloy Single Crystals with Explicit Microstructural Length Scales. Part II: Model Validation and Finite Element Implementation. Submitted for publication (2002).