Single-crystal Modeling

of Ni-based Superalloys

for Gas Turbine Blades

Adnan Hasanovic

Master thesis, MT 08.35

Supervisors:ir. Tiedo Tingadr. ir Marcel Brekelmansprof. dr. ir. Marc Geers

Mechanics of Materials,Department of Mechanical Engineering,Eindhoven University of Technology.September 29, 2008

Abstract

A recently proposed multiscale material model for single-crystal Ni-based su-peralloys is thoroughly tested and validated, mainly focusing on the effect ofchanges in microstructure on the mechanical response. This is done on both thematerial point microscopic level and the engineering (FE) macroscopic level. Formodeling, a computationally efficient unit cell approach is used that representsthe material’s microstructure. The unit cell is divided into a number of regionswith a certain volume. Quantities like stress and strain are computed per regionand the macroscopic stress and strain in a material point are obtained by takingthe volume average of the values of each individual region. Plastic deformationis calculated within a crystal plasticity framework. As the material consists oftwo different phases, interaction between these phases affects the macroscopicmaterial behavior and is modeled to account for phenomena that occur at theinterfaces. These phenomena are the lattice misfit and strain gradients. Whensubjected to a load at a sufficiently high temperature and stress, these typesof superalloys undergo microstructural geometrical changes that are known toinfluence the macroscopic material behavior as observed experimentally. Thedriving force for this so-called rafting effect is diffusion. An attempt is madeto model these microstructural changes phenomenologically and to investigatetheir effect on the macroscopic material behavior. Simulation results are quali-tatively compared to experimental data and reasonable agreement is found. Formore accurate modeling of rafting, however, the phenomenological model is notsufficient and physical deformation mechanisms such as precipitate shearing anddislocation climb need to be incorporated in the rafting model.

i

Contents

Abstract i

1 Introduction 1

2 Theoretical background 42.1 Multiscale modeling . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Coupling of different length scales . . . . . . . . . . . . . 82.1.3 Strain-gradient crystal plasticity . . . . . . . . . . . . . . 9

3 Simulation 123.1 Simulations at the material point level . . . . . . . . . . . . . . . 12

3.1.1 Uniaxial tension and uniaxial creep . . . . . . . . . . . . . 133.1.2 Size effects in uniaxial tension and uniaxial creep . . . . . 153.1.3 Volume fraction effects in uniaxial tension and uniaxial

creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.4 Comparison of simulation results with experiments . . . . 18

3.2 Simulations on the FE-level . . . . . . . . . . . . . . . . . . . . . 213.2.1 Creep simulation of the turbine blade . . . . . . . . . . . 223.2.2 Effect of size on the FE results . . . . . . . . . . . . . . . 243.2.3 Effect of precipitate volume fraction on the FE results . . 25

4 Rafting in the single-crystal superalloy CMSX-4 264.1 Origin of rafting in single-crystal superalloys . . . . . . . . . . . 264.2 Effect of rafting on the macroscopic material behavior . . . . . . 284.3 Modeling of rafting in single-crystal superalloys . . . . . . . . . . 29

4.3.1 Rafting model . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Modification of GND- and SSD-density . . . . . . . . . . 32

4.4 Simulation results with rafting effects . . . . . . . . . . . . . . . 324.4.1 Creep test simulation . . . . . . . . . . . . . . . . . . . . 334.4.2 Tensile test simulation . . . . . . . . . . . . . . . . . . . . 344.4.3 Simulation results compared to experiments . . . . . . . . 35

5 Conclusion 37

Bibliography 39

ii

Chapter 1

Introduction

The efficiency of gas turbines used in jet engines and industrial power plantsdepends, among others, highly on the operational temperature and rotationalspeed. Ability to increase both the (allowable) temperature and rotational speedresults in a more efficient turbine. This is especially of interest for the appli-cation of gas turbines in jet engines in the aerospace industry as with higheroperational temperatures and rotational velocities the possibility is created todesign smaller components as mass minimization is one of the important as-pects during the design process. In order to guarantee the structural integrityof the mechanical components, materials are used which exhibit the desired me-chanical properties at increased temperatures. For the design of gas turbinessingle-crystal Nickel-based (Ni) superalloys, such as CMSX-4, are widely used.The alloy CMSX-4 is known for its superior high temperature creep properties,which are a consequence of its two-phase microstructure. The microstructureconsists of a nickel matrix (γ) in which cubic Ni3Al (γ’) precipitates are dis-tributed. The distribution of the precipitates is more or less regular. The pre-cipitates have approximately constant size and shape (cubic) and are spaced atan approximately constant distance from each other. The volume fraction of theprecipitates is quite large, usually about 70 % in the commercially available al-loys. The microstructure of CMSX-4 is shown in figure 1.1. The microstructure

Figure 1.1: Microscopic image of the microstructure of CMSX-4 (left) and aschematic representation (right)

1

such as shown here is obtained by a controlled heat treatment. The commer-cially available CMSX-4 superalloy is designed such that the precipitates arecubic with an average dimension of 500 nm in the three orthogonal directions.The distance between the precipitates is approximately 60 nm.

For design and inspection purposes of mechanical components such as gasturbines, knowledge of the mechanical material behavior of single-crystal super-alloys is needed. Over the years significant effort has been made to characterizethe material behavior, using several techniques. The state of the material is pri-marily controlled by the amount of plastic deformation (high temperature creep)accumulated during the service time. On experimental basis, a lot of researchhas been done and main deformation mechanisms have been identified. In thefield of computational mechanics numerical tools have been developed whichare able to predict the material behavior up to a certain degree. However, thetwo-phase microstructure of single-crystal superalloys makes the developmentof proper material models somewhat difficult. The adoption of conventionalmacroscopic creep models is quite limited as these models lack the ability to ac-count for the microscopic parameters which determine the overall macroscopicbehavior. An example of such a microscopic parameter is the size (or shape) ofthe precipitate or the spacing between adjacent precipitates. These parametersare known to significantly influence the mechanical behavior of the material onmacroscopic level, as has been observed by tensile and creep experiments. Inorder to predict the influence of microscopic model parameters on the macro-scopic behavior, a numerical material model is required which accounts for theseeffects. In this study, results are presented using a recently developed multiscalematerial model which takes into account the microstructure effects. Due to itsmultiscale nature, the model accounts for microstructure effects and enables amechanical analysis on the macroscopic material point level which can furtherbe extended to the finite element (FE) level in a quite straightforward man-ner. For the material model to be suitable for an analysis in a finite elementenvironment, the computational efficiency is also an important aspect.

In the field of numerical modeling of the mechanical behavior of single-crystalNi-based superalloys, many difficulties still exist. Probably the most importantone is the fact that the mechanical properties change during the deformationprocess at certain conditions. This change in macroscopic mechanical propertiesis directly related to a change in the two-phase microstructure geometry of thealloy. This phenomenon is also referred to as rafting of the microstructure. Inorder for rafting to occur certain conditions need to be fulfilled, which is indeedthe case with a CMSX-4 alloy in service. These conditions are a relatively hightemperature and stress level (although the stress state plays a role too) andthe nature of the alloy. Single crystal Ni-based superalloys are so-called misfitalloys, where the misfit relates to the fact that the microstructure consists oftwo different phases which interact with each other at their contact interfaces.At these interfaces a certain geometrical misfit exists due to the different latticeparameters of both phases. This misfit leads to misfit strains and stresses,which affect the stress distribution in the material and thus play a role in thedeformation process in general and in the rafting process in particular. Therafting process has been modeled numerically but in none of the publishedliterature a model is presented which is able to describe rafting and which isalso applicable on the FE level at the same time.

The material model applied in this study is able to capture the effects of

2

rafting up to a certain degree both on material point level and the FE level.The applicability on FE level makes it possible to perform a mechanical analysisof complete structural components as a function of (service) time. For designand inspection purposes, the ability to predict the mechanical state of thesecomponents is highly desirable.

The global outline of this rapport is as follows. First, some theoreticalbackground and considerations are discussed which are the basis for the multi-scale constitutive model applied here. Then the general mechanical behavior ofthe model, presented in the form of tensile and creep simulations, is shown forvarious process conditions, such as temperature and deformation rate. Further,on the microscopic level parameters exist which affect the macroscopic materialbehavior. These are the matrix channel width (which provides a measure of theeffect of the length scale which is incorporated in the model) and precipitatevolume fraction. The effects of these parameters are investigated. The analysisis done on material point level and FE level. In the latter a complete turbineblade is analyzed. Numerical results are compared to experiments. In the firstpart of this study no rafting is taken into account. In the second part a raftingmodel is presented which extends the multiscale material model and makes itapplicable to simulate relatively high temperatures at which rafting occurs. Thenumerical results are analyzed and compared to experiments.

3

Chapter 2

Theoretical background

This chapter discusses briefly the basics of the theory used in the development ofthe multi-scale material model for the single-crystal Ni-based superalloy CMSX-4. A detailed description, however, is not provided as the model consists of alarge number of ingredients, which would result in a too extensive discussion.Nevertheless, to enhance the readability of this text, the reader is referred to(1) and (2), in which the full theoretical background is given.

In the following sections a short description is given of the various aspectsof the material model.

2.1 Multiscale modeling

In order to capture the two-phase nature of the material a multi-scale approachis adopted which accounts for different length scales. Three different lengthscales are distinguished, namely the macroscopic, mesoscopic and microscopiclength scale. The macroscopic length scale represents the engineering or FE-level. On mesoscopic level the microstructure is represented as a two-phase com-pound which exists of the nickel-based matrix (γ) and Ni3Al-based precipitates(γ′). The microscopic level is associated with the crystallographic response ofeach phase. This response is described using a rate-dependent crystal plasticityframework. The different length scales are schematically depicted in figure 2.1.

Figure 2.1: Multi-scale character of the material model

4

2.1.1 Unit cell

The mechanical behavor at a macroscopic material point is modeled using aunit cell approach. The microstructure and the unit cell are shown in figure 2.2and figure 2.3, respectively. The microstructure consists of a γ’-precipitateisurrounded by a γ-matrix. Between the matrix and the precipitate, inter-face regions are defined in which the effects occurring at the γ/γ’-interface areconcentrated. These interface regions behave either as the matrix or as the pre-cipitate with the difference that they account for the interfacial effects whichare illustrated further on in this section. The total number of different regionswithin the unit cell is 16. These regions are specified below:

� one precipitate (γ′) region

� three matrix (γ) channel regions (γj , j = 1...3) with different orientations(normal to the [001], [010] and [100]-direction)

� twelve interface regions (Ikm and Ikp, k = 1...6) which contain the inter-faces between the matrix and the precipitate

The interface regions are added in order to model the effects which take placeat γ/γ’-interfaces. These effects include the misfit stress and strain-gradientinduced back stress which are known to influence the macroscopic materialbehaviour. Misfit stress and strain-gradient induced back stress will be brieflydiscussed later on. The total interface regions consists of a precipitate interfaceregion (Ikp, k = 1...6) and a matrix interface region (Ikm, k = 1...6). Notethat the unit cell is periodic: the interface regions are found at all six faces ofthe cubic precipitate, whereas the matrix (bulk) regions are found only at threeof the six faces, see also figure 2.4. For the sake of computational efficiencyquantities like stress and strain are assumed to be uniform within a particularregion1. The overall unit cell response is obtained by volume averaging of thesecomponent-wise uniform quantities. The volume averaging procedure will bedescribed later on.

The microstructure morphology is defined by the geometrical parameters Li(i = 1,2,3), hi (i = 1,2,3), wim and wip (i = 1,...,6), see figure 2.2 and figure 2.3.The length parameter Li (i = 1,2,3) denotes the sum of the size of the precipitate(bulk region) and the width of the precipitate interface region. The width ofthe latter is defined as 5% of Li. It is assumed that the precipitate is cubic withLi equal in all three orthogonal directions. The parameter hi denotes the sumof the width of one matrix channel (bulk region) and the width of two matrixinterface regions. The width of one matrix interface region is assumed to be30% of hi (i = 1,2,3). As mentioned in Chapter 1, dimensions Li = 500 nm andhi = 60 nm give a precipitate volume fraction of 72%.

1Words ’phase’ and ’region’ are regularly used in this text. Note that ’phase’ refers to thematerial (either matrix or precipitate) while ’region’ refers to one of the building blocks of theunit cell (being either matrix of precipitate).

5

Figure 2.2: Microstructure in 2D

6

Figure 2.3: Unit cell in 2D

Figure 2.4: Unit cell in 3D

7

2.1.2 Coupling of different length scales

In order to relate the strain to the stress in a material point, different lengthscales need to be coupled. If one assumes that on the macroscopic level a certainamount of deformation (total strain) is prescribed, the stress is determined bycalculations on unit cell level (mesoscopic) and microscopic level using consti-tutive relations within a rate-dependent crystal plasticity framework. In thefollowing this procedure is briefly illustrated. For a detailed discussion, thereader is referred to (1).

Starting out with a known (prescribed) macroscopic deformation tensor, asystem of 120 equations can be constructed with stress and strain componentsin each region of the unit cell as variables to be solved. In this procedure onlyten of the sixteen regions (sixteen regions correspond to 192 unknown stress andstrain components) of the unit cell are considered: six interface regions on thefaces of the precipitate (see figure 2.4) are not modeled as these behave equallyin terms of stress and strain to the interface regions on the opposite side of theprecipitate. Instead, the volume of these regions is counted twice for volumeaveraging purposes, see details in (1). With the effective number of ten regions,the complete unit cell is involved with 60 strain and 60 stress quantities, makinga total of 120 unknowns. In each region, the stress and strain is assumed to beuniform. The (prescribed) macroscopic strain is the volume-average of strainsin all individual regions. For volume averaging purposes a modified Sachs ap-proach is used. This means that the stress in the matrix region and precipitateregion is uniform and equals the macroscopic stress. The averaged stress in thebi-crystals (pair of a matrix interface region and a precipitate interface region)is taken to be equal to the macroscopic stress. Further, kinematic compatibilitybetween the matrix interface regions and the precipitate interface regions needsto exist as both phases may respond differently to loading. Also, stress conti-nuity conditions exist at these interfaces: stresses acting on the separate phasesshould satisfy Newton’s action-reaction law.

The above considerations lead to a closed set of 120 equations which needto be solved for each increment in an incremental analysis.

8

2.1.3 Strain-gradient crystal plasticity

With stresses and strains known within each region (obtained by solving the setof 120 equations) of the unit cell, plastic strain rates can be calculated. Thesestrain rates are calculated on slip system level. Each region is assumed to dealwith 12 slip systems (FCC crystal). For each slip system, a plastic slip rate isdetermined. These individual plastic slip rates contribute to a total strain rateto be obtained for one specific region of the unit cell. When the strain ratesfor each region are known, an overall plastic strain rate for the complete unitcell is determined based on the volume average of strain rates for the individualregions. The overall plastic strain rate of the unit cell represents the plasticstrain rate for a material point (integration point in a FE environment).

For the calculation of plastic strain rates, the strain gradient effects are takeninto account. This gradient enhanced crystal plasticity approach is an exten-sion of conventional crystal plasticity. Although in traditional crystal plasticitythe plastic slip rate is determined on slip system level, the effect of individualdislocations is not considered. These dislocations are known to either supportthe plastic deformation or to impede it by their mutual interactions. Two typesof dislocations are distinguished:

� statistically stored dislocation (SSD’s)

� geometrically necessary dislocations (GND’s)

The first class of dislocations, SSD’s, are statistically distributed and ran-domly oriented and have effectively a zero Burgers vector. The second class,GND’s, is related to the heterogeneity of the plastic strain introduced in thematerial and causes a local stress field proportional to the strain gradients.As the strain field is usually not uniform (upon deformation) within a certainvolume of material, strain gradients exist. These strain gradients are accom-modated by a change in the density of GND’s. This change in GND density isnecessary to maintain the lattice compatibility upon deformation. Comparedto conventional crystal plasticity, the strain-gradient enhanced crystal plasticityallows to model the size effects which are related, apart from the Orowan stress,to the back stress. The back stress is related to the strain gradients, whichresult due to a nonuniform deformation field in the unit cell. Following thisline of reasoning, for example, a possible change in the matrix channel widthwill result in different strain gradients in the the matrix channel. The resultingbeck stress changes also and affects the macroscopic applied stress, implying theeffect of the length scale. These size effects are considered as the relatively nar-row matrix channels deform more easily than the precipitates, leading to largestrain gradients near the matrix-precipitate interface. For a detailed discussionof strain-gradient crystal plasticity, see (1) and (3).

The plastic strain rate for matrix channels and matrix interface regions iscalculated using the following slip law:

γα = γo

( |ταeff |sα

)m{1− exp

(−|ταeff |τor

)}nsign

(ταeff

), (2.1)

where ταeff and sα denote the effective resolved shear stress and the actual slipresistance (depending a.o. on temperature) on slip system α respectively and τor

9

is the Orowan stress. The parameters γ0, m and n are material constants. TheOrowan stress τor is a threshold for the effective stress that needs to be exceededin order to force individual dislocations between two adjacent precipitates, thusthrough a matrix channel. The Orowan stress threshold is determined by:

τor = aµb

d, (2.2)

with µ the shear modulus of the matrix, b the length of the Burgers vector and dthe spacing distance between two precipitates (thus the matrix channel width).The parameter a is a material constant.

The effective resolved shear stress ταeff depends on the slip direction and slipplane normal of the slip system. It also depends on the effective (unresolved)stress σeff which is the externally applied stress corrected with the stress fieldcaused by GND’s (see above) and the misfit stress. The latter is caused by thetwo-phase nature of the material.

The effective resolved shear stress ταeff is determined by:

ταeff = σeff : Pα, (2.3)

where Pα is the symmetric second order Schmid tensor defined as:

Pα =12

(~sα~nα + ~nα~sα), (2.4)

~nα and ~sα are, respectively, the slip plane normal and slip plane direction of slipsystem α. The second order stress tensor σeff in equation (2.3) is calculatedby:

σeff = σ + σmisfit − σback, (2.5)

with σ, σmisfit and σback the externally applied stresses, misfit stresses andback stresses, respectively. The tensors σmisfit and σback are also called inter-nal stresses as they are caused by dislocation effects. For a detailed discussion ofinternal stresses, slip resistance sα, GND’s and SSD’s and the exact calculationof these, see (1) and (3).

Equation (2.1) holds for the evolution of plastic strain in the matrix phase.A similar formulation exists for the precipitate phase, however in this case, alsothe effect of precipitate shearing and dislocation climb is taken into account,see also (2). Particle shearing can only occur at relatively high stress while fordislocation climb a relatively high temperature is a prerequisite. The slip lawfor the precipitate phase (both the bulk region and interface regions) is givenby:

γα =SbSintV

ραGNDmin

{fdiss

{1− exp

(−|ταeff |sα0

)}p+

4vαclimbHγ′

}sign

(ταeff

),

(2.6)

The parameter sα0 is the slip resistance on slip system α and is expected toremain constant for the precipitate phase as opposed to the matrix phase. In

10

the matrix phase, it is strongly controlled by the dislocation density, whereas itis expected that no significant changes of dislocation density occur in the pre-cipitate. The parameters S, Sint, V and Hγ′ are related to precipitate shearingand are expressed as functions of microstructural dimensions (Li and hi fori = 1, 2, 3) and b is the Burgers vector. The parameter fdiss is the dissociationfrequency and ραGNDmin

represents the number of available dislocation loops forslip system α. These latter two parameters are also related to particle shear-ing. The parameter vαclimb is the dislocation climb velocity and p is a materialconstant. For a more detailed discussion of the slip law for the matrix and pre-cipitate (equation (2.1) and equation (2.6)) and particularly the deformationmechanisms precipitate shearing and dislocation climb, see (1) and (2).

Finally, it is noted that for the elastic part of the deformation process or-thotropy with cubic symmetry is assumed. The elastic constants that determinethe stiffness matrix, as well as all other mentioned material constants, can befound in (1) and (2).

11

Chapter 3

Simulation

This chapter presents the simulation results obtained with the material modeldiscussed in the previous chapter. These results are compared to experimentalresults as far as the latter are available. The simulation is done on materialpoint level and engineering level (FE-level). While the material point analysisrelates the stress to a given strain in one single material point, the FE-analysisanalyzes a complete engineering component. In this case a turbine blade isconsidered. The rafting effect is not taken into account. This implies that themicrostructure morphology remains constant during the deformation process.Rafting effects are considered in the next chapter.

3.1 Simulations at the material point level

On material point level uniaxial tensile and uniaxial creep simulations are per-formed. First, simulation results are presented which show the general behaviourof the model as function of relevant (process) parameters. In case of a tensiletest simulation these parameters are temperature and strain rate. For creepsimulations, behavior is analyzed as function of temperature and applied stress.

The model applied in this study involves a large number of parameter. Thesemodel parameters can be divided in geometrical model parameters and materialconstants. The nmerical value of a large part of these parameters has beenmentioned in Chapter 2. Some of the parameters, however, are expressions ofother parameters which have not been mentioned in this study. Specification ofthe numerical values of these model parameters (i.e. fdiss or vαclimb) is more orless impossible while keeping the expressions of these parameters unmentioned inthis report. As already mentioned, the intention of these study is the sensitivityanalysis of the existing model and the extension to rafting. For a completepresentation of the existing material model and all model parameters, includingthe numerical values, see (1) and (1).

With respect to the material constants, the following comment is made.The model has been developed to cover a large range of operating conditionsfor a CMSX-4 alloy. The model will be applied in a temperature range of700-1100�. Analysis of creep should be possible for applied stresses up to 1GPa and the tensile behavior is to be analyzed for different strain rates. Inorder to cover all these operating conditions, the material constants have been

12

Figure 3.1: Tensile test at different strain rates and 800�

fitted to experiments performed at various conditions. The fitting procedurewas performed using the method of least squares. For more information on thematerial constant fitting procedure, see again (1) and (1).

3.1.1 Uniaxial tension and uniaxial creep

The uniaxial tensile simulations are strain-controlled in the loading direction:a constant strain rate is prescribed within a time increment. In the transversedirection, the two stress components are required to be equal to zero. In uni-axial creep simulations the total initial stress is applied within the first timeincrement. Figure 3.1 shows the stress-strain curves for different strain ratesat a temperature of 800�. Due to the visco-plastic nature of the model, therate-dependent behavior is clearly observed. In figure 3.2 results are shown atdifferent temperatures and constant strain rate. The effect of an increasing tem-perature on the slip resistance is clearly visible in a lower initial yield stress.Also the steady-state flow stress, defined as the more or less stabilized stresslevel which is reached after some 3 % total strain, decreases as the temperatureincreases. Figure 3.3 shows the accumulated creep strain at a constant temper-ature and different levels of the applied stress. In figure 3.4 the creep strainis shown at a constant applied stress and different temperatures. Figure 3.3shows what is expected: higher stresses result in more plastic deformation. Infigure 3.4 again the effect of increased temperature is observed in larger plasticdeformation due to the decreased slip resistance of the material.

13

Figure 3.2: Tensile test at different temperatures (in�) and strain rate of 0.0011/s

Figure 3.3: Creep test at different levels of the applied stress and 800�

14

Figure 3.4: Creep test at constant stress of 250 MPa and different temperatures(in�)

3.1.2 Size effects in uniaxial tension and uniaxial creep

Experimental studies clearly show a size dependency of the mechanical responseof the superalloy CMSX-4. This size dependency is the result of the lengthscale effect due to the microstructure’s two-phase nature and morphology. Inthe previous chapter it has been explained that the microstructure consistsof precipitates distributed in a nickel matrix. The precipitates are spaced atcertain distance apart, which is the matrix channel width. Due to the lower slipresistance of the matrix and the interaction between the matrix and precipitateat their interfaces, upon loading the matrix will deform more easily than theprecipitate. The reasons for this difference are the ability of dislocations tomove relatively easily through the matrix and their inability to intrude theprecipitate. This intrusion of the precipitate is caused by the dissociation inpartial dislocations, a mechanism which occurs only if certain conditions arefulfilled, see (2). Another mechanism that controls the plastic deformation isdislocation climb, however, in order for it to occur, a relatively high temperatureis required as dislocation climb is a diffusion related process, see again (2).

As mentioned earlier, the length scale effect is incorporated in the materialmodel by the Orowan stress treshold which needs to be overcome by the act-ing stress. Also, the strain-gradient effects are responsible for a length scaleeffect. In order to analyze the model’s sensitivity to the length scale effect interms of macroscopic response, the unit cell size is decreased or increased bya factor. This factor scales all dimensions (of all regions) within the unit cell.For example, a factor of 0.50 reduces the dimensions of all regions with 50 %.

15

Figure 3.5: Size effect with 0.001 1/s and 800�

The shape (ratio) of different regions remains thus unchanged. This reductionof dimensions results a.o. in a decrease of the matrix channel width. As theplastic deformation is primarily carried by moving dislocations through the ma-trix material, this results in an impeded dislocation movement, as bowing adislocation between two adjacent precipitates becomes more difficult. This cor-responds to an increased Orowan threshold which requires an increased stresslevel to keep the dislocation activity sufficiently high. Furthermore, the straingradients increase due to a smaller channel width which results in a higher backstress. This back stress affects the stress field in the material. These effectsresult in relatively less plastic deformation of the matrix1.

The stress-strain curves with different unit cell sizes are shown in figure 3.5.The 100 % (red) line corresponds to the reference microstructure geometry, thatis, Li = 500 nm and hi = 60 nm for i = 1,2,3. Figure 3.6 shows the creep curves.These curves show that less plastic deformation occurs with smaller unit cellsizes. This is clearly visible on both the increased initial yield stress and stresslevels in the plastic region of the curves.

1For simplicity reasons, here it is implicitly assumed that the stress and temperature arenot sufficiently high in order to initiate precipitate shearing and dislocation climb, althoughthe statement holds also in case of plastic deformation of the precipitate

16

Figure 3.6: Size effect with 400 MPa and 800�

3.1.3 Volume fraction effects in uniaxial tension and uni-axial creep

Next to the size of different regions within the unit cell, the microstructuremorphology controls the volume fraction of the two phases as well. Althoughno direct comparison with experimental data is possible (due to the lack ofexperiments) it is interesting to take a look at the simulation results for differentprecipitate volume fractions. Here, the dimensions of the precipitate, Li (i =1,2,3), see figure 2.3, are varied. The width of the precipitate interface regions,wpi (i = 1,...6), changes also as it defined as 5 % of Li. The matrix channelwidth, hi (i = 1,2,3), is kept constant at 60 nm. Also, the width of the matrixchannel interface region, wmi (i = 1,...6), remains constant as it is defined as30 % of hi. These modifications in the microstructure morphology result in achange of the volume fraction of γ′. The precipitate shape is kept unchanged.

Figure 3.7 and figure 3.8 show the stress-strain curves and creep curves,respectively. The creep curves indicate that less plastic deformation occurs forlarger precipitate volume fractions. This is not surprising as the slip resistance ofa precipitate is higher than the slip resistance of the matrix. A relatively largerportion of the unit cell consisting of precipitate material results thus in lessplastic deformation for an unchanged applied stress and temperature. However,there is also another, somewhat less obvious way in which the volume fractionof γ′ may affect the macroscopic response. As explained in Chapter 2, the widthof the precipitate interface regions, wpi (i = 1,...6), is coupled to the precipitatedimensions Li (i = 1,2,3). Increasing the precipitate dimensions will thus leadto an increase of the width of the precipitate interface regions and thus decreasethe strain gradients. Smaller strain gradients will lead to a lower back stressinduced by dislocations and may thus affect the effective resolved shear stress onthe slip system level. A potentially increased effective resolved shear stress will

17

Figure 3.7: Precipitate volume fraction effect with 0.001 1/s and 800�

produce larger plastic strain rates in certain regions en thus result in an overallincreased plastic deformation. Following this line of reasoning, the effect of theprecipitate volume fraction is determined by two counteracting effects: the slipresistance of γ′ and the strain gradients. The net resulting effect is thus difficultto predict. Although the creep curves clearly show that the plastic deformationdecreases with a higher precipitate volume fraction (figure 3.8), the hardeningeffect is hardly visible in the stress-strain curves (figure 3.7).

3.1.4 Comparison of simulation results with experiments

In this paragraph the simulation results are compared to the experimental data.The amount of available experimental data is, however, rather limited. Experi-mental tensile data, including the size effects on tensile behavior, are availableand are thus presented in the following. Experimental data with different pre-cipitate volume fractions, however, is not available, as the precipitate volumefraction of commercial superalloys is usually approximately 0.7. Figure 3.9shows four curves. Two of them are experimentally obtained with a differentmicrostructure morphology at a temperature of 950� and a strain rate of 0.0011/s. In the production process of superalloys a different morphology is realizedby a different heat treatment. The other two curves are predicted by the model.Both the experimental curves and predicted curves show more or less the sametrend: for a microstructure with a smaller size, thus the 384/57 nm curve (Li =384 nm and hi = 57 nm, i = 1,2,3), the macroscopic material response is stifferthan for the case with larger microstructural dimensions of 500/60 nm (Li = 500nm and hi = 60 nm, i = 1,2,3). In figure 3.9 it can be seen that the simulationresults deviate from the experimental data. One of possible explanations for

18

Figure 3.8: Precipitate volume fraction effect with 400 MPa and 800�

this difference may be the fact that the material constants have been fitted tothe behavior from a large range of testing conditions and that the model is notable to capture the behavior at one specific testing condition very accurately.

The effect of size can also be seen in figure 3.10. Here, the normalized steady-state flow stress is plotted as function of the relative size. The experimental datais obtained at 850� and 0.001 1/s. The normalized steady-state flow stress isthe more or less stabilized stress level which is reached after approximately 3%strain. The relative size is a factor by which the unit cell dimensions are scaledin order to obtain either smaller or larger sizes of different regions within theunit cell. The size effect is clearly seen both in the experimental data and thesimulation results, although the size effect in the simulations is somewhat lesspronounced than in the experiments. The reason for this slight deviation maybe the fact that the experimental data is obtained with a superalloy (PWA1480)very similar to CMSX-4 but not identical. This is because the experimental datafor CMSX-4 was not available. Nevertheless, the comparison shows a quite goodagreement between experimental data and simulation results.

19

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

1

2

3

4

5

6

7

8

9

10x 10

8

Strain [%]

Str

ess

[Pa]

simulation 500/60 nm

experiment 500/60 nm

simulation 384/57 nm

experiment 384/57 nm

Figure 3.9: Comparison of simulation results with experimental data for uniaxialtension at 950� and 0.001 1/s

0.6 0.8 1 1.2 1.4 1.60.9

0.95

1

1.05

1.1

1.15

Relative size [−]

Nor

mal

ized

ste

ady−

stat

e flo

w s

tres

s [−

]

experimentsimulation

Figure 3.10: Comparison of simulation results compared to experimental datafor uniaxial tension at 850� and 0.001 1/s

20

3.2 Simulations on the FE-level

This section presents the macroscopic FE-simulations in which the material be-havior is described by the multi-scale material model applied above. The resultsthat were shown in previous sections are on material point level, which is equiv-alent to the integration point level within a FE framework. In this section, aFE mesh is considered of one single turbine blade. The mesh consists of 5463linear hexahedrons and 7302 nodes, see figure 3.11. The z-displacement of thelower side of the base of the blade is constrained. Additional constraints are ap-plied in x- and y-direction in order to suppress rigid body motion, thus avoidinga singular stiffness matrix. A centrifugal load equivalent to 10 000 revolutionsper minute is applied. Further, there is a pressure difference of 0.5 bar overthe blade thickness as the result of the aerodynamic load due to the expansionof the gas in the turbine. The temperature distribution over the blade is notuniform due to the nonuniform temperature of the gas exiting the combustionchamber. As the material behavior depends on the temperature, more (plastic)deformation will result at locations with a higher temperature. The tempera-ture varies between 900� and 950�. The temperature distribution has beendetermined by a steady-state thermal analysis and is shown in figure 3.12.

Figure 3.11: FE-model of the turbine blade

21

Figure 3.12: Temperature distribution � in the turbine blade

3.2.1 Creep simulation of the turbine blade

Creep simulations are performed in which the turbine blade is supposed to besubjected to constant loading. The creep analysis is done for rather short creeptimes due to numerical issues. Nevertheless, the simulation provides insightin the mechanical behavior of the turbine, especially when the effect of themicrostructure morphology is taken into account, see below.

The overall plastic deformation is shown in figure 3.13. The results in fig-ure 3.13 are obtained with the microstructure as initially found in commercialsuperalloys. This means that the precipitate dimensions of 500 nm and matrixchannel width of 60 nm are used. The maximum creep strains are found nearthe base of the blade. As can be observed, the plastic deformation is calculatedonly in the areas where relatively high stresses exist. During the elasto-plasticanalysis, the plastic strains were not calculated in the low stressed areas as theplastic deformation here is relatively small compared to plastic deformation inhighly stressed areas. These relatively small values of plastic strain are notconsidered relevant as one of the primary goals of this FE-analysis is to identifylocations with significant plastic deformation. The choice not to run the multi-scale model for the complete turbine blade allows much shorter run-times. Thestress distribution is shown in figure 3.14. The highest stresses are found nearthe base of the turbine blade.

22

Figure 3.13: Plastic deformation in the turbine blade

Figure 3.14: Distribution of the equivalent stress [Pa] in the turbine blade

23

3.2.2 Effect of size on the FE results

Creep simulations are also performed with different microstructure morpholo-gies. This paragraph shows results for microstructures with different sizes ofregions within the unit cell. All sizes are scaled with a certain factor, analo-gously to the simulations at material point level in section 3.1.2. Figure 3.15shows the creep strain as a function of time for various microstructures, againat the location where the maximum plastic deformation (node 5151, see fig-ure 3.11) is found.

0 5 10 15 20 25 300

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time [s]

Equ

ival

ent c

reep

str

ain

[−]

65 %

100 %

200 %

Figure 3.15: Plastic deformation for various sizes of the microstructures. Thecreep strain data is extracted at node 5151, the location of the maximum plasticdeformation

In figure 3.15 it can be seen that a smaller microstructural size reveals a stifferresponse, as was the case in the material point analysis.

24

3.2.3 Effect of precipitate volume fraction on the FE re-sults

This paragraph shows the effect of the precipitate volume fraction on the FEresults. Analogous to the variation of the precipitate volume fraction in section3.1.3, also here the volume of the precipitate is changed and its effects on theturbine blade is analyzed. Results are shown in figure 3.16. As can be seen infigure 3.16. The result is comparable to the result found at the material pointlevel.

0 5 10 15 20 250

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Time [s]

Equ

ival

ent c

reep

str

ain

[−]

vf = 0.55

vf = 0.71

vf = 0.84

Figure 3.16: Plastic deformation for various precipitate volume fractions. Thecreep strain data is extracted at node 5151, the location of the maximum plasticdeformation

In figure 3.16 it can be seen that larger γ′ volume fractions result in a stifferresponse, as was the case in the material point analysis.

25

Chapter 4

Rafting in the single-crystalsuperalloy CMSX-4

The simulation results presented so far do not account for the so-called raftingeffect. Rafting is microstructural adaptation that occurs in single-crystal super-alloys at temperatures above 950�, see (4) to (9). For the full characterizationof the mechanical behavior of single-crystal superalloys, modeling of the raft-ing phenomenon is necessary, as the rafts start to develop early in the loadingprocess and affect the macroscopic behavior of the material. This suggests thatthe applicability of the material model dealt with in the previous sections islimited to low and moderate temperatures (< 950�). Even at these lower tem-peratures rafting will occur if the material is subjected to load for a sufficientlylong period. These conditions are typical for the steady-state operation of gasturbines in service and in order to be able to predict its mechanical behaviorafter longer service times, modeling of rafting is required.

In the following section, the notion and origin of rafting in single-crystalsuperalloys is explained. In the remaining sections of this chapter, a numericalmodel is presented, which takes into account the effects of rafting, and simulationresults are discussed.

4.1 Origin of rafting in single-crystal superalloys

Rafting in single-crystal superalloys occurs as a consequence of high temper-ature, stress level and stress state, the two-phase nature of the material andtime during which the material is subjected to load (4), (5), (6) and (7). Thedriving force for rafting is related to the diffusion of atoms from one phase tothe other and vice versa, depending on the location within the microstructure.The progression of microstructure rafting can be seen in figure 4.1. Initially,the precipitates are cubic measuring 500 x 500 x 500 nm and the horizontalmatrix channel width is approximately 60 nm. After a certain time and undera certain stress (state) and temperature, the microstructure has evolved to theintermediate state with partly developed rafts. The precipitates extend in adirection perpendicular to the loading axis and become flatter. They tend totake the shape of thin flat plates. Also, but less clearly visible in figure 4.1, thewidth of the matrix channels perpendicular to the loading axis increases, called

26

Figure 4.1: Various stages of rafting in single-crystal superalloy CMSX-4

channel widening. Partly, the γ′-phase is replaced with the γ-phase at somelocations and the opposite happens at other locations. In the fully rafted state,thus when two adjacent precipitates meet, the microstructure can be thought ofas a lamellar structure of both phases. The time after which the rafting processis completed depends on the temperature, the stress and the lattice parameterof both phases. The lattice parameter is important as it causes a certain misfitbetween the phases. In CMSX-4 the misfit is called negative which means thatthe lattice constant of the precipitates phase is slightly smaller than the latticeconstant of the matrix phase. The misfit interferes with the externally appliedstress and initiates diffusion of atoms between the two phases. This is explainedin a little bit more detail using figure 4.2. In the left part of figure 4.2, where no

Figure 4.2: Schematic illustration of the effects of various stress componentson rafting in the single-crystal superalloy CMSX-4. Stress-free state (left) andloaded state (right)

external stress is applied, the only stresses present are the misfit stresses foundat the interfaces of both phases. The chemical potentials of both phases at point1 and 2 (γ′ in the lower right of the left figure) are equal. Under these circum-

27

stances, due to the presence of misfit stress, ’rafting’ will occur in all directionat an equal rate (also called nondirectional coarsening). In the right-hand sideof the figure, however, a tensile stress is applied. The resulting stresses at loca-tions 1 and 2 are not equal in this case. The local stresses in both phases aremodified in the direction of the external stress. In the direction perpendicular tothe external stress axis stresses change due to the different Poisson numbers ofthe matrix and precipitate, however, this effect is small. All stress componentscombine to yield one specific hydrostatic stress σh (σh = 1

3 (σ1 + σ2 + σ3)).This hydrostatic stress is known to directly influence the chemical potentials ofatoms. Tensile stresses act in the γ′-phase parallel to the γ-channels. Theseare balanced by the compressive stresses in the γ-channels parallel to the γ′-surfaces which are much larger than small tensile stress components which actperpendicular to the channels.

Rafting is driven by the tendency of the system to decrease its overall γ′/γ-interface energy. Therefore σh and thus the chemical potential of atoms whichis directly influenced by σh, see (10), are different. Differences in chemicalpotentials of atoms are known to act as driving force for diffusion (11). Tosummarize, the chemical potentials of the atoms at locations 1 and 2 whichwere equal in case of zero external stress are not longer equal in the case whenthe external stress is nonzero. This stress difference leads to a driving force fordiffusion and results finally in the observed rafting effect. The net effect is thusthe diffusion of γ′-atoms from the top and bottom surface of the precipitate(thus near the interface with horizontal matrix channels) towards the verticalmatrix channels, thereby filling up the space which was initially filled withmatrix phase. As the precipitate atoms move from the top and bottom surfaceof the precipitate, this space is filled with the matrix atoms from the horizontalchannels. Effectively, the width of the horizontal channels is thus increased andthe width of the vertical matrix channels is decreased.

4.2 Effect of rafting on the macroscopic materialbehavior

The effect of rafting in single-crystal superalloys has remained not well under-stood during a quite long period. Experimental studies showed controversialresults until it was recognized that the effect of rafting could not be assessed fora general set of testing conditions but rather for specific temperature and stressranges. One of the explanations for the increasing creep behavior as the result ofrafting was based on the assumption that dislocation glide, which is one of thedeformation mechanisms, is eliminated due to the lamellar structure of precipi-tate and matrix phase. Further deformation should occur by the mechanism ofprecipitate shearing, which requires stresses to be sufficiently high. For stressesbelow 150 MPa, precipitate shearing is most likely not to occur even at rela-tively high temperatures. The result is a higher resistance to deformation andincreased creep behavior. On the other hand, when the temperature and stressare relatively high (> 950� and > 150 MPa, respectively), the relative escaperate of interfacial dislocations is reduced so the dislocation path is decreasedand also the stress is high enough to enable precipitate shearing, deformation ismore easily realized. This results in a deteriorated macroscopic creep behavior.

28

In order to show the effect of rafting on the creep behavior, virgin and pre-raftedsamples have been used to perform experiments at different testing conditions.The results are shown in figure 4.3.

Figure 4.3: Comparison of creep rupture of single-crystal superalloys with avirgin (cubes) and rafted microstructre (Source: (4))

From figure 4.3 it follows that at relatively low temperatures (950�) and highstresses the creep rupture time is shorter for samples with a rafted microstruc-ture: rupture is attained at a lower stress than for samples without rafts forequal loading times. The same is true at relatively high temperature of 1050�for relatively high stresses. Only at stresses below 150 MPa the samples with arafted microstructure exhibit longer creep resistance, that is, a higher stress isnecessary for material rupture to occur after a certain loading time, comparedto the material with a virgin microstructure.

4.3 Modeling of rafting in single-crystal super-alloys

In this section a model is represented that describes rafting of the unit cell.

4.3.1 Rafting model

In order to model rafting effects on the macroscopic material behavior, descrip-tion of the microstructure morphology change is necessary. In reality the rateand degree of rafting in CMSX-4 will depend on temperature, stress and time.Some authors report that rafting also depends on accumulated creep strain ata certain instant in time. For the description of geometrical changes in themicrostructure morphology, a model is presented in this section. The model is

29

represented by an evolution equation which relates the rate of change of the pre-cipitate dimensions to the temperature and stress. For simplicity, the possibleeffect of the present creep strain is not taken into account. Figure 4.4 illus-trates the geometrical changes that occur during rafting. As already explained,the horizontal matrix channels widen, while the vertical channels shrink due todiffusion of precipitate atoms. As precipitate atoms move towards the verti-cal matrix channels, the precipitate changes its shape from a cube to a plate.In case of a one-dimensional stress state, the normal on the plate structure isoriented parallel to the stress axis. The dimensions in the unit cell are related

Figure 4.4: Geometrical changes in microstructure morphology due to rafting(2D)

to each other by the equations shown below. This thus means that knowledgeof the new precipitate shape determines the dimensions of the matrix channelregions. The volume invariance of the precipitate is assumed, as well as volumeinvariance of the matrix. A set of 6 equations is constructed which yields allnew dimensions (Li+1

1 , L1+12 , Li+1

3 , hi+11 , hi+1

2 and hi+13 ) in the unit cell.

The evolution equations for the precipitate are a modification of the equationpresented in (12). The modifications are made in order to account for a multi-dimensional stress state. Two evolution equations for in three directions aredefined:

L1(T, σ) = −32L1

[ 23σ11 − 1

3σ22 − 13σ33

σVM

]A

L· exp

[−Q− U(T ) · σVM

RT

](4.1)

L2(T, σ) = −32L2

[ 23σ22 − 1

3σ11 − 13σ33

σVM

]A

L· exp

[−Q− U(T ) · σVM

RT

](4.2)

L3(T, σ) = −32L3

[ 23σ33 − 1

3σ11 − 13σ22

σVM

]A

L· exp

[−Q− U(T ) · σVM

RT

](4.3)

where A/L is a pre-exponential factor, Q the activation energy, R the universalgas constant and U(T ) the temperature dependent activation volume described

30

by a power function:

U(T ) = UT (T − T0)n (4.4)

where T > T0 = 1100.7 K. UT is a constant.

Further, σii (i = 1, 2, 3) are the stress components in the three orthogonaldirections and σVM is the equivalent Von Mises stress. The fitted constantsA/L = 186.3 · 103 h−1, Q = 221780 J/mol, UT = 0.19 J/ (mol · MPa · Kn) andn = 1.294 are independent of temperature and stress.

The equations (4.1) - (4.3) imply volume-invariance of the precipitate:

Vγ′ = L1L2L3 = constant (4.5)

After the precipitate dimensions have been determined, the matrix channelwidths are calculated with following equations:

h1 = C − L1 (4.6)

h2 = C − L2 (4.7)

h3 = C − L3 (4.8)

with C the constant equal to 560 · 10−9 m. which is the size of the complete unitcell in the all three directions. The latter three equations assure that the shapeof the unit cell remains unchanged, in other words the precipitate is allowed toreshape at cost of the matrix and vice versa. The fact that C is a constant,implies that the model does not account for microstructure coarsening, which isthe shape change of the unit cell that occurs in the late stages of the degradationprocess.

As mentioned, equations (4.1) - (4.3) are a slight modification of the equa-tion presented in (12). This latter equation is found to reasonably describe theevolution of the width of horizontal channels in an uniaxial stress state for awide range of temperatures, stresses and loading times, see also (12). In the im-plementation the above rafting equations are evaluated at each time increment.

Next to the formation of a lamellar structure of matrix and precipitate phase,which is here referred to as microstructure rafting, also microstructure coarsen-ing can occur in single-crystal superalloys, see (12). Microstructure coarseningoccurs after the lamellar structure is fully developed. The time necessary forcoarsening highly depends on the temperature and stress. During the coarsen-ing the width of the horizontal matrix channels keeps increasing, although at astrongly decreasing rate, see also (12). For simplicity reasons, this effect is nottaken into account in the model discussed in this thesis, although it could beincorporated relatively easy by defining an evolution equation for C in equa-tions (4.6) - (4.8). The omission of coarsening effects is one of the limitationsof this rafting model.

31

4.3.2 Modification of GND- and SSD-density

When no rafting occurs, the volume of each separate region does not changeand the SSD-density in each region is defined as the number of dislocations perunit volume. However, when rafting occurs, the volume of individual regions(bulk and interface regions) is allowed to change, even though the total volumeof both the matrix phase and the precipitate phase is conserved. During thegeometry evolution process, the total number of GND’s as well as SSD’s isassumed to be constant for each individual region of the unit cell. Variation ofthe volume of an individual region thus affects the dislocation density in thisregion. As the dislocation density is defined as the number of dislocations perunit volume, the dislocation density in a particular region is corrected with thevolume ratio (of the same region) between two subsequent increments. In thisway the SSD-density in a particular region will proportionally increase with thevolume reduction of this region. In case the volume increases, the SSD-densitywill decrease. Thus, the following relation between the SSD-density and volumeratio of two subsequent increments for each region is valid:

ρSSDi+1 = ρSSDi /J (4.9)

where ρSSDi is the SSD-density in the previous increment, ρSSDi+1 the SSD-densityin the current increment and J the volume ratio of the specific region.

The GND-density in the interface regions is determined by the slip gradients.The width of the interface regions thus directly influences the GND-density inthe interface region. A correction step is performed that takes into account thechange of the GND-density in the interface regions due to the change of theinterface region width. This is achieved with the following relations:

ρGNDi+1 = ρGNDcorrectedi + ∆ρ (4.10)

where ρGNDcorrectedi is the (due to changed slip gradients) corrected GND-density

in the previous increment, defined as:

ρGNDcorrectedi = ρGNDi

wnewwold

(4.11)

with wold and wnew the channel width in two subsequent time increments. TheGND-density in the matrix and precipitate bulk regions remains unchanged.

4.4 Simulation results with rafting effects

The simulation results presented in chapter 3 do not take the rafting effect intoaccount. The microstructure morphology remains unchanged: the precipitateretains its initial shape of a cube and the matrix channel width remains inde-pendent of temperature and stress as the deformation progresses. This resultsin an overestimated mechanical response at the macroscopic level while experi-mental data clearly reveal a clear deterioration in the mechanical behavior, seefor example (12).

32

In this section, simulation results including the rafting effect are presentedand compared to simulation data generated with no rafting taken into account.The calculated results are also compared to experiments. In order to show themodel capabilities only results at material point level are presented. The exten-sion to a FE-environment is rather straight forward. The results on materialpoint level are presented for uniaxial tension and creep. Before uniaxial tensilesimulations are performed, creep simulations are done in order to arrive at arafted microstructure, so the material has indeed a pre-rafted microstructure atthe instant that the tensile test begins. For creep simulations, the microstruc-ture naturally evolves during the loading, as loading times are much longer thanin case of the tensile tests.

4.4.1 Creep test simulation

Figure 4.5 shows the creep behavior at different temperature and stresses withand without rafting. The effect of a changing microstructure morphology isclearly present. More plastic deformation is observed when rafting effects aretaken into account.

0 2 4 6 8 10

x 105

0

0.5

1

1.5x 10

−3

Time [s]

Cre

ep s

trai

n [%

]

No raftingRafting

(a) Temperature = 900� and σ = 150 MPa

0 2 4 6

x 105

0

0.005

0.01

0.015

0.02

0.025

0.03

Time [s]

Cre

ep s

trai

n [%

]

No raftingRafting

(b) Temperature = 900� and σ = 300 MPa

0 0.5 1 1.5 2

x 105

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

Time [s]

Cre

ep s

trai

n [%

]

No raftingRafting

(c) Temperature = 1000� and σ = 150 MPa

0 2 4 6

x 104

0

0.01

0.02

0.03

0.04

0.05

0.06

Time [s]

Cre

ep s

trai

n [%

]

No raftingRafting

(d) Temperature = 1000� and σ = 300 MPa

Figure 4.5: Effect of rafting on the creep behavior at different temperature andstress levels

33

4.4.2 Tensile test simulation

In this paragraph the rafting effect on the tensile behavior is shown. Resultsincluding the rafting effect are generated after a creep simulation has been per-formed, which assures that the microstructure is pre-rafted at the instant atwhich the simulation of the tensile test is started. The tensile test is simu-lated at 900� with a strain rate of 0.001 1/s. The results are shown with un-rafted, partially rafted and fully rafted (lamellar) microstructure. Pre-raftingwas achieved at a temperature of 1000� and a stress of 300 MPa for 35 000s (’partially rafted’) and 70 000 s (’fully rafted’). In case of a fully rafted mi-crostructure, the horizontal matrix channel width has increased from 60 nm to160 nm. The vertical channel widths are equal to zero (adjacent precipitatesmeet). The stress-strain curves are shown in figure 4.6.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9

10x 10

8

Strain [%]

Str

ess

[Pa]

Unrafted

Partially rafted

Fully rafted

Figure 4.6: Effect of rafting on the tensile behavior at a temperature of 900�and strain rate of 0.001 1/s. The microstructure is pre-rafted at a temperatureof 1000� and a stress of 300 MPa for 35 000 s (’partially rafted’) and 70 000 s(’fully rafted’)

Figure 4.6 clearly shows a softening effect of rafting on the tensile behavior.Although the yield stress is hardly affected, the post-yielding behavior showssignificant dependence on (the degree of) rafting.

34

4.4.3 Simulation results compared to experiments

In this section a qualitative comparison between the simulation results andexperiments is discussed. The experimental data available is based on stress-strain data of tensile bars which have been pre-rafted prior to the tensile test.The comparison between simulation results and experimental data is shown infigure 4.7. During the experiment the material was pre-rafted in a creep test at1050� with 68 MPa during 2500 hours. Tensile bars with this microstructurewere then tested in uniaxial tension at 900� and a strain rate of 0.0006 1/s. Inthe numerical analysis the rafting conditions were simulated before a uniaxialtensile analysis was performed. Also shown in figure 4.7 is the simulated curvewithout rafting effects. Although the simulation results deviate somewhat fromthe experimental data, the figure clearly shows that the simulation includingthe rafting model approaches the experimental curve better than the simulationwithout rafting. When the rafting model is incorporated in the simulation, asoftening effect is clearly visible, which is also observed experimentally ((4),(12)). The deviation between the simulated and experimental results is mostprobably caused by the fact that at these pre-rafting conditions also coarseningoccurs, see also (12). In fact, the reported channel width after degradation was348 nm, while in the simulation the channel width has reached a maximum of160 nm. Coarsening degrades the microstructure even further which results inan even more softened behavior. This is very likely as the experimental curvepredicts a lower stress than the simulated curve.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

10x 10

8

Strain [%]

Str

ess

[Pa]

Simulation (rafted microstructure)

Experiment (rafted microstructure)

Simulation (unrafted microstructure)

Figure 4.7: Comparison between simulation results and experimental data oftensile bars with rafted microstructure at 900� and a strain rate of 0.0006 1/s

35

Another possible explanation for the deviating results is the fact that therafting model does not account for the effect of a rafted unit cell on variousdeformation mechanisms in a rafted microstructure. It is very likely that pre-cipitate shearing and dislocation climb (2) are operating somewhat differently ina rafted microstructure compared to a virgin microstructure. Due to a differentshape of the precipitate, it is expected that the dislocation movement aroundthe precipitate is affected, which in turn affects the macroscopic plastic deforma-tion level. The phenomenological modeling of rafting as presented in this study(which is taking into account only the geometrical changes of the microstructureand strain gradients but none of the physical deformation mechanisms) is prob-ably not sufficient for capturing all rafting effects in a rafted microstructure veryaccurately. This is also shown in figure 4.8 where stress-strain data is shown forrafted microstructures which are not coarsened, as microstructure coarseningstarts only for h3 > 160 nm. This implies that the deviation of the simulationresults is not only due to the fact that microstructure coarsening is not con-sidered, but also may be attributed to changes in the deformation mechanisms,which have not been included in the rafting model. Finally, note that also thetensile test for the virgin material is not simulated very accurately, which im-plies that the model parameter set used is not optimal for the test conditionsused here, see also figure 3.9, which shows a comparison between the simulationresults and experiments for materials with a virgin (unrafted) microstructure.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

9

10x 10

8

Strain [%]

Str

ess

[Pa]

Simulation: Virgin

Simulation: h3 = 86 nm

Simulation: h3 = 160 nm

Experiment: Virgin

Experiment: h3 = 86 nm

Experimentn: h3 = 160 nm

Figure 4.8: Comparison between simulation results and experimental data oftensile bars with rafted microstructure at 950� and strain rate of 0.001 1/s

36

Chapter 5

Conclusion

In this study a multi-scale material model for the single-crystal Ni-based su-peralloy CMSX-4 has been dicussed and the macroscopic material behavior wasanalyzed in dependence on the microstructure morphology. The effect of thesize of microstructural dimensions was investigated, as well as the effect of theprecipitate volume fraction. It was found that smaller unit cell dimensions(preserving the volume ratio of precipitate and matrix) lead to a hardening ef-fect. In creep simulations, significantly less plastic deformation is found for thesmaller sizes. This is also the case in simulations of tensile tests, where boththe initial yield stress and steady-state flow stress are found to be higher forsmaller microstructure sizes. A hardening effect, although somewhat less pro-nounced, was also found for higher precipitate volume fractions. Results werecompared with experimental data and reasonable agreement was found, see fig-ures 3.9, 4.7 and 4.8. The deviations from the experimental data is believedto be caused by the fact the material constants were fitted to a large amount ofdata which cover large ranges of operating conditions. Probably, more accuratesimulation results would have been obtained by fitting the constants to datacovering narrower ranges of working conditions, however, this would somewhatlimit the model’s applicability. Simulation results for different precipitate vol-ume fractions could, however, not be compared to experimental data as the alloyCMSX-4 is currently commercially available only with a γ′-volume fraction of0.72. The model was successfully used in a FE environment which allows themechanical analysis of complete turbine blades.

For temperatures above 950� a rafting model was proposed which sim-ulates the microstructure evolution quite reasonably. The simulation resultswere compared to experimental stress-strain data obtained on pre-rafted tensilebars. The degree of rafting depends on the temperature, stress and loadingtime. This dependence is also observed in the experiments. The rafting modelas presented in this study is only able to describe the rafting process which isthe evolution of the microstructure until the instant at which two adjacent pre-cipitates meet. Although in reality coarsening occurs after the rafting process iscompleted, the microstructure coarsening is not covered by the current raftingmodel. Nevertheless, the trend due to microstructure degeneration, which is thematerial softening as observed in experiments, is clearly predicted by the mate-rial model. It is believed that the somewhat deviating numerical results from theexperimental data are due to the omission of coarsening effects. Furthermore,

37

for more accurate modeling of rafting effects, the phenomenological model isnot sufficient as it lacks the incorporation of the effects of physical deformationmechanisms, such as precipitate shearing and dislocation climb. It is believedthat these two mechanisms are active when rafting occurs and that their effectsare different in a rafted unit cell compared to a virgin unit cell. Extension of therafting model to account correctly for the effect of the deformation mechanismmay significantly improve the accuracy of the simulation.

Although the presented model including the rafting effects is also applicablein a FE environment, FE-results were not presented. The model can however beused in the FE analysis of a complete turbine blade in a straightforward man-ner, similarly to the case where rafting effects were neglected, see also Chapter3, paragraph 3.2. In this way, the FE-analysis may indicate the degree of mi-crostructure degeneration at specific locations in the FE-model (i.e. turbineblade) as a function of the temperature, stress (state) and service time of thecomponent.

38

Bibliography

[1] T. Tinga, W.A.M. Brekelmans, M.G.D. Geers, Incorporating Strain-Gradient Effects in a Multi-Scale Framework for Nickel-base Superalloys,Philosophical Magazine, available online, 2008

[2] T. Tinga, W.A.M. Brekelmans, M.G.D. Geers, Micro-mechanical Consti-tutive Model for a Nickel-base Superalloy, personal communication with T.Tinga

[3] L. P. Evers, D. M. Parks, W. A. M. Brekelmans and M. G. D. Geers,Crystal plasticity model with enhanced hardening by geometrically necessarydislocation accumulation, Journal of the Mechanics and Physics of Solids,Volume 50, Issue 11, 2002, pp. 2403-2424

[4] M. Kamaraj, Rafting in Single-Crystal Nickel-base Superalloys – AnOverview, Sadhana, Vol. 28, Parts 1 & 2, 2003, pp. 115-128

[5] N. Matan, D. C. Cox, C. M. F. Rae, R. C. Reed, On the Kinetics of Raftingin CMSX-4 Superalloy Single-Crystals, Acta Materialia, Volume 47, Issue7, 1999, pp. 2031-2045

[6] R. C. Reed, N. Matan, D.C. Cox, M. A. Rist, C. M. F. Rae, Creep of CMSX-4 Superalloy Single-Crystals: Effects of Rafting at High Temperature, ActaMaterialia , Volume 47, Issue 12, 1999, pp. 3367-3381

[7] K. Serin, G. Gobenli, G. Eggeler, On the Influence of Stress State, StressLevel and Temperature on γ-Channel Widening in the Single-Crystal Su-peralloy CMSX-4, ICSMA 13 International Conference on the Strength ofMaterials No13, Budapest, Hungary, 2004, vol. 387-89, pp. 133-137

[8] M. Kamaraj, K. Serin, M. Kolbe, G. Eggeler, Influence of Stress Stateon the Kinetics of γ-Channel Widening during High Temperature and LowStress Creep of the Single-Crystal Speralloy CMSX-4, CSMA-12: Inter-national Conference on the Strength of Materials No12, Asilomar, CA,ETATS-UNIS, 2001, vol. 319/21, pp. 796-799

[9] P. Henderson, L. Berglin, C. Jansson, On Rafting in a Single-CrystalNickel-base Superalloy after High and Low Temperature Creep, Scripta Ma-terialia, Volume 40, Number 2, 1998, pp. 229-234

[10] R. W. Swalin, Thermodynamics of Solids, John Wiley and Sons, 1967

[11] P. G. Shewmon, Diffusion in Solids, Warrendale, PA Miner., Met. & Mater.Soc., 1989

39

[12] A. Epishin, T. Link, M. Nazmy, M. Staubli, H. Klingelhoffer, G. Nolze, Mi-crostructural Degradation of CMSX-4: Kinetics and Effect on MechanicalProperties, Superalloys 2008

40