Single-crystal Modeling of Ni-based Superalloys for Gas Turbine Blades

Transcription

1 Single-crystal Modeling of Ni-based Superalloys for Gas Turbine Blades Adnan Hasanovic Master thesis, MT Supervisors: ir. Tiedo Tinga dr. ir Marcel Brekelmans prof. dr. ir. Marc Geers Mechanics of Materials, Department of Mechanical Engineering, Eindhoven University of Technology. September 29, 2008

2 Abstract A recently proposed multiscale material model for single-crystal Ni-based superalloys is thoroughly tested and validated, mainly focusing on the effect of changes in microstructure on the mechanical response. This is done on both the material point microscopic level and the engineering (FE) macroscopic level. For modeling, a computationally efficient unit cell approach is used that represents the material s microstructure. The unit cell is divided into a number of regions with a certain volume. Quantities like stress and strain are computed per region and the macroscopic stress and strain in a material point are obtained by taking the volume average of the values of each individual region. Plastic deformation is calculated within a crystal plasticity framework. As the material consists of two different phases, interaction between these phases affects the macroscopic material behavior and is modeled to account for phenomena that occur at the interfaces. These phenomena are the lattice misfit and strain gradients. When subjected to a load at a sufficiently high temperature and stress, these types of superalloys undergo microstructural geometrical changes that are known to influence the macroscopic material behavior as observed experimentally. The driving force for this so-called rafting effect is diffusion. An attempt is made to model these microstructural changes phenomenologically and to investigate their effect on the macroscopic material behavior. Simulation results are qualitatively compared to experimental data and reasonable agreement is found. For more accurate modeling of rafting, however, the phenomenological model is not sufficient and physical deformation mechanisms such as precipitate shearing and dislocation climb need to be incorporated in the rafting model. i

3 Contents Abstract i 1 Introduction 1 2 Theoretical background Multiscale modeling Unit cell Coupling of different length scales Strain-gradient crystal plasticity Simulation Simulations at the material point level Uniaxial tension and uniaxial creep Size effects in uniaxial tension and uniaxial creep Volume fraction effects in uniaxial tension and uniaxial creep Comparison of simulation results with experiments Simulations on the FE-level Creep simulation of the turbine blade Effect of size on the FE results Effect of precipitate volume fraction on the FE results Rafting in the single-crystal superalloy CMSX Origin of rafting in single-crystal superalloys Effect of rafting on the macroscopic material behavior Modeling of rafting in single-crystal superalloys Rafting model Modification of GND- and SSD-density Simulation results with rafting effects Creep test simulation Tensile test simulation Simulation results compared to experiments Conclusion 37 Bibliography 39 ii

4 Chapter 1 Introduction The efficiency of gas turbines used in jet engines and industrial power plants depends, among others, highly on the operational temperature and rotational speed. Ability to increase both the (allowable) temperature and rotational speed results in a more efficient turbine. This is especially of interest for the application of gas turbines in jet engines in the aerospace industry as with higher operational temperatures and rotational velocities the possibility is created to design smaller components as mass minimization is one of the important aspects during the design process. In order to guarantee the structural integrity of the mechanical components, materials are used which exhibit the desired mechanical properties at increased temperatures. For the design of gas turbines single-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 microstructure consists of a nickel matrix (γ) in which cubic Ni 3 Al (γ ) precipitates are distributed. The distribution of the precipitates is more or less regular. The precipitates have approximately constant size and shape (cubic) and are spaced at an approximately constant distance from each other. The volume fraction of the precipitates is quite large, usually about 70 % in the commercially available alloys. 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 a schematic representation (right) 1

5 such as shown here is obtained by a controlled heat treatment. The commercially available CMSX-4 superalloy is designed such that the precipitates are cubic 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 gas turbines, knowledge of the mechanical material behavior of single-crystal superalloys is needed. Over the years significant effort has been made to characterize the material behavior, using several techniques. The state of the material is primarily controlled by the amount of plastic deformation (high temperature creep) accumulated during the service time. On experimental basis, a lot of research has been done and main deformation mechanisms have been identified. In the field of computational mechanics numerical tools have been developed which are able to predict the material behavior up to a certain degree. However, the two-phase microstructure of single-crystal superalloys makes the development of proper material models somewhat difficult. The adoption of conventional macroscopic creep models is quite limited as these models lack the ability to account for the microscopic parameters which determine the overall macroscopic behavior. An example of such a microscopic parameter is the size (or shape) of the precipitate or the spacing between adjacent precipitates. These parameters are known to significantly influence the mechanical behavior of the material on macroscopic level, as has been observed by tensile and creep experiments. In order to predict the influence of microscopic model parameters on the macroscopic behavior, a numerical material model is required which accounts for these effects. In this study, results are presented using a recently developed multiscale material model which takes into account the microstructure effects. Due to its multiscale nature, the model accounts for microstructure effects and enables a mechanical analysis on the macroscopic material point level which can further be extended to the finite element (FE) level in a quite straightforward manner. For the material model to be suitable for an analysis in a finite element environment, the computational efficiency is also an important aspect. In the field of numerical modeling of the mechanical behavior of single-crystal Ni-based superalloys, many difficulties still exist. Probably the most important one is the fact that the mechanical properties change during the deformation process at certain conditions. This change in macroscopic mechanical properties is directly related to a change in the two-phase microstructure geometry of the alloy. This phenomenon is also referred to as rafting of the microstructure. In order for rafting to occur certain conditions need to be fulfilled, which is indeed the case with a CMSX-4 alloy in service. These conditions are a relatively high temperature and stress level (although the stress state plays a role too) and the nature of the alloy. Single crystal Ni-based superalloys are so-called misfit alloys, where the misfit relates to the fact that the microstructure consists of two different phases which interact with each other at their contact interfaces. At these interfaces a certain geometrical misfit exists due to the different lattice parameters 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 the deformation process in general and in the rafting process in particular. The rafting process has been modeled numerically but in none of the published literature a model is presented which is able to describe rafting and which is also applicable on the FE level at the same time. The material model applied in this study is able to capture the effects of 2

6 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 analysis of complete structural components as a function of (service) time. For design and inspection purposes, the ability to predict the mechanical state of these components is highly desirable. The global outline of this rapport is as follows. First, some theoretical background and considerations are discussed which are the basis for the multiscale constitutive model applied here. Then the general mechanical behavior of the model, presented in the form of tensile and creep simulations, is shown for various process conditions, such as temperature and deformation rate. Further, on the microscopic level parameters exist which affect the macroscopic material behavior. These are the matrix channel width (which provides a measure of the effect of the length scale which is incorporated in the model) and precipitate volume fraction. The effects of these parameters are investigated. The analysis is done on material point level and FE level. In the latter a complete turbine blade is analyzed. Numerical results are compared to experiments. In the first part of this study no rafting is taken into account. In the second part a rafting model is presented which extends the multiscale material model and makes it applicable to simulate relatively high temperatures at which rafting occurs. The numerical results are analyzed and compared to experiments. 3

7 Chapter 2 Theoretical background This chapter discusses briefly the basics of the theory used in the development of the 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 a large 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 aspects of the material model. 2.1 Multiscale modeling In order to capture the two-phase nature of the material a multi-scale approach is adopted which accounts for different length scales. Three different length scales are distinguished, namely the macroscopic, mesoscopic and microscopic length scale. The macroscopic length scale represents the engineering or FElevel. On mesoscopic level the microstructure is represented as a two-phase compound which exists of the nickel-based matrix (γ) and Ni 3 Al-based precipitates (γ ). The microscopic level is associated with the crystallographic response of each phase. This response is described using a rate-dependent crystal plasticity framework. The different length scales are schematically depicted in figure 2.1. Figure 2.1: Multi-scale character of the material model 4

8 2.1.1 Unit cell The mechanical behavor at a macroscopic material point is modeled using a unit cell approach. The microstructure and the unit cell are shown in figure 2.2 and figure 2.3, respectively. The microstructure consists of a γ -precipitate isurrounded by a γ-matrix. Between the matrix and the precipitate, interface regions are defined in which the effects occurring at the γ/γ -interface are concentrated. These interface regions behave either as the matrix or as the precipitate with the difference that they account for the interfacial effects which are illustrated further on in this section. The total number of different regions within the unit cell is 16. These regions are specified below: ˆ one precipitate (γ ) region ˆ three matrix (γ) channel regions (γ j, j = 1...3) with different orientations (normal to the [001], [010] and [100]-direction) ˆ twelve interface regions (I k m and I k p, k = 1...6) which contain the interfaces between the matrix and the precipitate The interface regions are added in order to model the effects which take place at γ/γ -interfaces. These effects include the misfit stress and strain-gradient induced back stress which are known to influence the macroscopic material behaviour. Misfit stress and strain-gradient induced back stress will be briefly discussed later on. The total interface regions consists of a precipitate interface region (I p k, k = 1...6) and a matrix interface region (I m k, k = 1...6). Note that the unit cell is periodic: the interface regions are found at all six faces of the cubic precipitate, whereas the matrix (bulk) regions are found only at three of the six faces, see also figure 2.4. For the sake of computational efficiency quantities like stress and strain are assumed to be uniform within a particular region 1. The overall unit cell response is obtained by volume averaging of these component-wise uniform quantities. The volume averaging procedure will be described later on. The microstructure morphology is defined by the geometrical parameters L i (i = 1,2,3), h i (i = 1,2,3), w m i and w p i (i = 1,...,6), see figure 2.2 and figure 2.3. The length parameter L i (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 of the latter is defined as 5% of L i. It is assumed that the precipitate is cubic with L i equal in all three orthogonal directions. The parameter h i denotes the sum of the width of one matrix channel (bulk region) and the width of two matrix interface regions. The width of one matrix interface region is assumed to be 30% of h i (i = 1,2,3). As mentioned in Chapter 1, dimensions L i = 500 nm and h i = 60 nm give a precipitate volume fraction of 72%. 1 Words phase and region are regularly used in this text. Note that phase refers to the material (either matrix or precipitate) while region refers to one of the building blocks of the unit cell (being either matrix of precipitate). 5

9 Figure 2.2: Microstructure in 2D 6

10 Figure 2.3: Unit cell in 2D Figure 2.4: Unit cell in 3D 7

11 2.1.2 Coupling of different length scales In order to relate the strain to the stress in a material point, different length scales need to be coupled. If one assumes that on the macroscopic level a certain amount of deformation (total strain) is prescribed, the stress is determined by calculations on unit cell level (mesoscopic) and microscopic level using constitutive relations within a rate-dependent crystal plasticity framework. In the following this procedure is briefly illustrated. For a detailed discussion, the reader is referred to (1). Starting out with a known (prescribed) macroscopic deformation tensor, a system of 120 equations can be constructed with stress and strain components in each region of the unit cell as variables to be solved. In this procedure only ten of the sixteen regions (sixteen regions correspond to 192 unknown stress and strain components) of the unit cell are considered: six interface regions on the faces of the precipitate (see figure 2.4) are not modeled as these behave equally in terms of stress and strain to the interface regions on the opposite side of the precipitate. Instead, the volume of these regions is counted twice for volume averaging 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, making a total of 120 unknowns. In each region, the stress and strain is assumed to be uniform. The (prescribed) macroscopic strain is the volume-average of strains in all individual regions. For volume averaging purposes a modified Sachs approach is used. This means that the stress in the matrix region and precipitate region is uniform and equals the macroscopic stress. The averaged stress in the bi-crystals (pair of a matrix interface region and a precipitate interface region) is taken to be equal to the macroscopic stress. Further, kinematic compatibility between the matrix interface regions and the precipitate interface regions needs to exist as both phases may respond differently to loading. Also, stress continuity conditions exist at these interfaces: stresses acting on the separate phases should satisfy Newton s action-reaction law. The above considerations lead to a closed set of 120 equations which need to be solved for each increment in an incremental analysis. 8

12 2.1.3 Strain-gradient crystal plasticity With stresses and strains known within each region (obtained by solving the set of 120 equations) of the unit cell, plastic strain rates can be calculated. These strain rates are calculated on slip system level. Each region is assumed to deal with 12 slip systems (FCC crystal). For each slip system, a plastic slip rate is determined. These individual plastic slip rates contribute to a total strain rate to be obtained for one specific region of the unit cell. When the strain rates for each region are known, an overall plastic strain rate for the complete unit cell is determined based on the volume average of strain rates for the individual regions. The overall plastic strain rate of the unit cell represents the plastic strain rate for a material point (integration point in a FE environment). For the calculation of plastic strain rates, the strain gradient effects are taken into account. This gradient enhanced crystal plasticity approach is an extension of conventional crystal plasticity. Although in traditional crystal plasticity the plastic slip rate is determined on slip system level, the effect of individual dislocations is not considered. These dislocations are known to either support the plastic deformation or to impede it by their mutual interactions. Two types of dislocations are distinguished: ˆ statistically stored dislocation (SSD s) ˆ geometrically necessary dislocations (GND s) The first class of dislocations, SSD s, are statistically distributed and randomly oriented and have effectively a zero Burgers vector. The second class, GND s, is related to the heterogeneity of the plastic strain introduced in the material and causes a local stress field proportional to the strain gradients. As the strain field is usually not uniform (upon deformation) within a certain volume of material, strain gradients exist. These strain gradients are accommodated by a change in the density of GND s. This change in GND density is necessary to maintain the lattice compatibility upon deformation. Compared to conventional crystal plasticity, the strain-gradient enhanced crystal plasticity allows 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, which result due to a nonuniform deformation field in the unit cell. Following this line of reasoning, for example, a possible change in the matrix channel width will result in different strain gradients in the the matrix channel. The resulting beck stress changes also and affects the macroscopic applied stress, implying the effect of the length scale. These size effects are considered as the relatively narrow matrix channels deform more easily than the precipitates, leading to large strain gradients near the matrix-precipitate interface. For a detailed discussion of strain-gradient crystal plasticity, see (1) and (3). The plastic strain rate for matrix channels and matrix interface regions is calculated using the following slip law: γ α = γ o ( τ α eff s α ) m { 1 exp ( τ α eff τ or )} n sign ( τ α eff ), (2.1) where τ α eff and sα denote the effective resolved shear stress and the actual slip resistance (depending a.o. on temperature) on slip system α respectively and τ or 9

13 is the Orowan stress. The parameters γ 0, m and n are material constants. The Orowan stress τ or is a threshold for the effective stress that needs to be exceeded in order to force individual dislocations between two adjacent precipitates, thus through a matrix channel. The Orowan stress threshold is determined by: τ or = a µb d, (2.2) with µ the shear modulus of the matrix, b the length of the Burgers vector and d the spacing distance between two precipitates (thus the matrix channel width). The parameter a is a material constant. The effective resolved shear stress τeff α depends on the slip direction and slip plane normal of the slip system. It also depends on the effective (unresolved) stress σ eff which is the externally applied stress corrected with the stress field caused by GND s (see above) and the misfit stress. The latter is caused by the two-phase nature of the material. The effective resolved shear stress τ α eff is determined by: τ α eff = σ eff : P α, (2.3) where P α is the symmetric second order Schmid tensor defined as: P α = 1 2 ( sα n α + n α s α ), (2.4) n α and s α are, respectively, the slip plane normal and slip plane direction of slip system α. The second order stress tensor σ eff in equation (2.3) is calculated by: σ eff = σ + σ misfit σ back, (2.5) with σ, σ misfit and σ back the externally applied stresses, misfit stresses and back stresses, respectively. The tensors σ misfit and σ back are also called internal stresses as they are caused by dislocation effects. For a detailed discussion of internal stresses, slip resistance s α, GND s and SSD s and the exact calculation of 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, also the effect of precipitate shearing and dislocation climb is taken into account, see also (2). Particle shearing can only occur at relatively high stress while for dislocation climb a relatively high temperature is a prerequisite. The slip law for the precipitate phase (both the bulk region and interface regions) is given by: { γ α = SbS ( int ρ α GND V min f diss {1 exp τ eff α )} p } s α + 4vα climb sign ( τ α ) eff, 0 H γ (2.6) The parameter s α 0 is the slip resistance on slip system α and is expected to remain constant for the precipitate phase as opposed to the matrix phase. In 10

14 the matrix phase, it is strongly controlled by the dislocation density, whereas it is expected that no significant changes of dislocation density occur in the precipitate. The parameters S, S int, V and H γ are related to precipitate shearing and are expressed as functions of microstructural dimensions (L i and h i for i = 1, 2, 3) and b is the Burgers vector. The parameter f diss is the dissociation frequency and ρ α GND min represents the number of available dislocation loops for slip system α. These latter two parameters are also related to particle shearing. The parameter vclimb α is the dislocation climb velocity and p is a material constant. For a more detailed discussion of the slip law for the matrix and precipitate (equation (2.1) and equation (2.6)) and particularly the deformation mechanisms precipitate shearing and dislocation climb, see (1) and (2). Finally, it is noted that for the elastic part of the deformation process orthotropy with cubic symmetry is assumed. The elastic constants that determine the stiffness matrix, as well as all other mentioned material constants, can be found in (1) and (2). 11

15 Chapter 3 Simulation This chapter presents the simulation results obtained with the material model discussed in the previous chapter. These results are compared to experimental results as far as the latter are available. The simulation is done on material point level and engineering level (FE-level). While the material point analysis relates the stress to a given strain in one single material point, the FE-analysis analyzes a complete engineering component. In this case a turbine blade is considered. The rafting effect is not taken into account. This implies that the microstructure 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 performed. First, simulation results are presented which show the general behaviour of the model as function of relevant (process) parameters. In case of a tensile test simulation these parameters are temperature and strain rate. For creep simulations, behavior is analyzed as function of temperature and applied stress. The model applied in this study involves a large number of parameter. These model parameters can be divided in geometrical model parameters and material constants. The nmerical value of a large part of these parameters has been mentioned in Chapter 2. Some of the parameters, however, are expressions of other parameters which have not been mentioned in this study. Specification of the numerical values of these model parameters (i.e. f diss or vclimb α ) is more or less impossible while keeping the expressions of these parameters unmentioned in this report. As already mentioned, the intention of these study is the sensitivity analysis of the existing model and the extension to rafting. For a complete presentation of the existing material model and all model parameters, including the 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 conditions for a CMSX-4 alloy. The model will be applied in a temperature range of Analysis of creep should be possible for applied stresses up to 1 GPa and the tensile behavior is to be analyzed for different strain rates. In order to cover all these operating conditions, the material constants have been 12

16 Figure 3.1: Tensile test at different strain rates and 800 fitted to experiments performed at various conditions. The fitting procedure was performed using the method of least squares. For more information on the material constant fitting procedure, see again (1) and (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 transverse direction, the two stress components are required to be equal to zero. In uniaxial creep simulations the total initial stress is applied within the first time increment. Figure 3.1 shows the stress-strain curves for different strain rates at a temperature of 800. Due to the visco-plastic nature of the model, the rate-dependent behavior is clearly observed. In figure 3.2 results are shown at different temperatures and constant strain rate. The effect of an increasing temperature 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 stress level which is reached after some 3 % total strain, decreases as the temperature increases. Figure 3.3 shows the accumulated creep strain at a constant temperature and different levels of the applied stress. In figure 3.4 the creep strain is shown at a constant applied stress and different temperatures. Figure 3.3 shows what is expected: higher stresses result in more plastic deformation. In figure 3.4 again the effect of increased temperature is observed in larger plastic deformation due to the decreased slip resistance of the material. 13

17 Figure 3.2: Tensile test at different temperatures (in ) and strain rate of /s Figure 3.3: Creep test at different levels of the applied stress and

18 Figure 3.4: Creep test at constant stress of 250 MPa and different temperatures (in ) Size effects in uniaxial tension and uniaxial creep Experimental studies clearly show a size dependency of the mechanical response of the superalloy CMSX-4. This size dependency is the result of the length scale effect due to the microstructure s two-phase nature and morphology. In the previous chapter it has been explained that the microstructure consists of precipitates distributed in a nickel matrix. The precipitates are spaced at certain distance apart, which is the matrix channel width. Due to the lower slip resistance of the matrix and the interaction between the matrix and precipitate at their interfaces, upon loading the matrix will deform more easily than the precipitate. The reasons for this difference are the ability of dislocations to move relatively easily through the matrix and their inability to intrude the precipitate. This intrusion of the precipitate is caused by the dissociation in partial dislocations, a mechanism which occurs only if certain conditions are fulfilled, see (2). Another mechanism that controls the plastic deformation is dislocation climb, however, in order for it to occur, a relatively high temperature is required as dislocation climb is a diffusion related process, see again (2). As mentioned earlier, the length scale effect is incorporated in the material model by the Orowan stress treshold which needs to be overcome by the acting stress. Also, the strain-gradient effects are responsible for a length scale effect. In order to analyze the model s sensitivity to the length scale effect in terms of macroscopic response, the unit cell size is decreased or increased by a 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

19 Figure 3.5: Size effect with /s and 800 The shape (ratio) of different regions remains thus unchanged. This reduction of dimensions results a.o. in a decrease of the matrix channel width. As the plastic deformation is primarily carried by moving dislocations through the matrix material, this results in an impeded dislocation movement, as bowing a dislocation between two adjacent precipitates becomes more difficult. This corresponds to an increased Orowan threshold which requires an increased stress level to keep the dislocation activity sufficiently high. Furthermore, the strain gradients increase due to a smaller channel width which results in a higher back stress. This back stress affects the stress field in the material. These effects result in relatively less plastic deformation of the matrix 1. 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, that is, L i = 500 nm and h i = 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 cell sizes. This is clearly visible on both the increased initial yield stress and stress levels in the plastic region of the curves. 1 For simplicity reasons, here it is implicitly assumed that the stress and temperature are not sufficiently high in order to initiate precipitate shearing and dislocation climb, although the statement holds also in case of plastic deformation of the precipitate 16

20 Figure 3.6: Size effect with 400 MPa and Volume fraction effects in uniaxial tension and uniaxial creep Next to the size of different regions within the unit cell, the microstructure morphology controls the volume fraction of the two phases as well. Although no direct comparison with experimental data is possible (due to the lack of experiments) it is interesting to take a look at the simulation results for different precipitate volume fractions. Here, the dimensions of the precipitate, L i (i = 1,2,3), see figure 2.3, are varied. The width of the precipitate interface regions, w p i (i = 1,...6), changes also as it defined as 5 % of L i. The matrix channel width, h i (i = 1,2,3), is kept constant at 60 nm. Also, the width of the matrix channel interface region, wi m (i = 1,...6), remains constant as it is defined as 30 % of h i. These modifications in the microstructure morphology result in a change 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 for larger precipitate volume fractions. This is not surprising as the slip resistance of a precipitate is higher than the slip resistance of the matrix. A relatively larger portion of the unit cell consisting of precipitate material results thus in less plastic deformation for an unchanged applied stress and temperature. However, there is also another, somewhat less obvious way in which the volume fraction of γ may affect the macroscopic response. As explained in Chapter 2, the width of the precipitate interface regions, w p i (i = 1,...6), is coupled to the precipitate dimensions L i (i = 1,2,3). Increasing the precipitate dimensions will thus lead to an increase of the width of the precipitate interface regions and thus decrease the strain gradients. Smaller strain gradients will lead to a lower back stress induced by dislocations and may thus affect the effective resolved shear stress on the slip system level. A potentially increased effective resolved shear stress will 17

21 Figure 3.7: Precipitate volume fraction effect with /s and 800 produce larger plastic strain rates in certain regions en thus result in an overall increased plastic deformation. Following this line of reasoning, the effect of the precipitate volume fraction is determined by two counteracting effects: the slip resistance of γ and the strain gradients. The net resulting effect is thus difficult to predict. Although the creep curves clearly show that the plastic deformation decreases with a higher precipitate volume fraction (figure 3.8), the hardening effect is hardly visible in the stress-strain curves (figure 3.7) 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. Experimental tensile data, including the size effects on tensile behavior, are available and are thus presented in the following. Experimental data with different precipitate volume fractions, however, is not available, as the precipitate volume fraction of commercial superalloys is usually approximately 0.7. Figure 3.9 shows four curves. Two of them are experimentally obtained with a different microstructure morphology at a temperature of 950 and a strain rate of /s. In the production process of superalloys a different morphology is realized by 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 same trend: for a microstructure with a smaller size, thus the 384/57 nm curve (L i = 384 nm and h i = 57 nm, i = 1,2,3), the macroscopic material response is stiffer than for the case with larger microstructural dimensions of 500/60 nm (L i = 500 nm and h i = 60 nm, i = 1,2,3). In figure 3.9 it can be seen that the simulation results deviate from the experimental data. One of possible explanations for 18

22 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 to the behavior from a large range of testing conditions and that the model is not able to capture the behavior at one specific testing condition very accurately. The effect of size can also be seen in figure Here, the normalized steadystate flow stress is plotted as function of the relative size. The experimental data is obtained at 850 and /s. The normalized steady-state flow stress is the 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 scaled in order to obtain either smaller or larger sizes of different regions within the unit cell. The size effect is clearly seen both in the experimental data and the simulation results, although the size effect in the simulations is somewhat less pronounced than in the experiments. The reason for this slight deviation may be 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 data for CMSX-4 was not available. Nevertheless, the comparison shows a quite good agreement between experimental data and simulation results. 19

23 10 x Stress [Pa] simulation 500/60 nm experiment 500/60 nm simulation 384/57 nm experiment 384/57 nm Strain [%] Figure 3.9: Comparison of simulation results with experimental data for uniaxial tension at 950 and /s 1.15 experiment simulation Normalized steady state flow stress [ ] Relative size [ ] Figure 3.10: Comparison of simulation results compared to experimental data for uniaxial tension at 850 and /s 20

24 3.2 Simulations on the FE-level This section presents the macroscopic FE-simulations in which the material behavior is described by the multi-scale material model applied above. The results that were shown in previous sections are on material point level, which is equivalent to the integration point level within a FE framework. In this section, a FE mesh is considered of one single turbine blade. The mesh consists of 5463 linear hexahedrons and 7302 nodes, see figure The z-displacement of the lower side of the base of the blade is constrained. Additional constraints are applied in x- and y-direction in order to suppress rigid body motion, thus avoiding a singular stiffness matrix. A centrifugal load equivalent to revolutions per minute is applied. Further, there is a pressure difference of 0.5 bar over the blade thickness as the result of the aerodynamic load due to the expansion of the gas in the turbine. The temperature distribution over the blade is not uniform due to the nonuniform temperature of the gas exiting the combustion chamber. As the material behavior depends on the temperature, more (plastic) deformation will result at locations with a higher temperature. The temperature varies between 900 and 950. The temperature distribution has been determined by a steady-state thermal analysis and is shown in figure Figure 3.11: FE-model of the turbine blade 21

25 Figure 3.12: Temperature distribution in the turbine blade Creep simulation of the turbine blade Creep simulations are performed in which the turbine blade is supposed to be subjected to constant loading. The creep analysis is done for rather short creep times due to numerical issues. Nevertheless, the simulation provides insight in the mechanical behavior of the turbine, especially when the effect of the microstructure morphology is taken into account, see below. The overall plastic deformation is shown in figure The results in figure 3.13 are obtained with the microstructure as initially found in commercial superalloys. This means that the precipitate dimensions of 500 nm and matrix channel width of 60 nm are used. The maximum creep strains are found near the base of the blade. As can be observed, the plastic deformation is calculated only in the areas where relatively high stresses exist. During the elasto-plastic analysis, the plastic strains were not calculated in the low stressed areas as the plastic deformation here is relatively small compared to plastic deformation in highly stressed areas. These relatively small values of plastic strain are not considered relevant as one of the primary goals of this FE-analysis is to identify locations with significant plastic deformation. The choice not to run the multiscale model for the complete turbine blade allows much shorter run-times. The stress distribution is shown in figure The highest stresses are found near the base of the turbine blade. 22

26 Figure 3.13: Plastic deformation in the turbine blade Figure 3.14: Distribution of the equivalent stress [Pa] in the turbine blade 23

27 3.2.2 Effect of size on the FE results Creep simulations are also performed with different microstructure morphologies. This paragraph shows results for microstructures with different sizes of regions within the unit cell. All sizes are scaled with a certain factor, analogously to the simulations at material point level in section Figure 3.15 shows the creep strain as a function of time for various microstructures, again at the location where the maximum plastic deformation (node 5151, see figure 3.11) is found Equivalent creep strain [ ] % 100 % 200 % Time [s] Figure 3.15: Plastic deformation for various sizes of the microstructures. The creep strain data is extracted at node 5151, the location of the maximum plastic deformation In figure 3.15 it can be seen that a smaller microstructural size reveals a stiffer response, as was the case in the material point analysis. 24

28 3.2.3 Effect of precipitate volume fraction on the FE results This paragraph shows the effect of the precipitate volume fraction on the FE results. Analogous to the variation of the precipitate volume fraction in section 3.1.3, also here the volume of the precipitate is changed and its effects on the turbine blade is analyzed. Results are shown in figure As can be seen in figure The result is comparable to the result found at the material point level Equivalent creep strain [ ] vf = 0.55 vf = 0.71 vf = Time [s] Figure 3.16: Plastic deformation for various precipitate volume fractions. The creep strain data is extracted at node 5151, the location of the maximum plastic deformation In figure 3.16 it can be seen that larger γ volume fractions result in a stiffer response, as was the case in the material point analysis. 25

29 Chapter 4 Rafting in the single-crystal superalloy CMSX-4 The simulation results presented so far do not account for the so-called rafting effect. Rafting is microstructural adaptation that occurs in single-crystal superalloys at temperatures above 950, see (4) to (9). For the full characterization of the mechanical behavior of single-crystal superalloys, modeling of the rafting phenomenon is necessary, as the rafts start to develop early in the loading process and affect the macroscopic behavior of the material. This suggests that the applicability of the material model dealt with in the previous sections is limited to low and moderate temperatures (< 950 ). Even at these lower temperatures rafting will occur if the material is subjected to load for a sufficiently long period. These conditions are typical for the steady-state operation of gas turbines in service and in order to be able to predict its mechanical behavior after longer service times, modeling of rafting is required. In the following section, the notion and origin of rafting in single-crystal superalloys is explained. In the remaining sections of this chapter, a numerical model is presented, which takes into account the effects of rafting, and simulation results are discussed. 4.1 Origin of rafting in single-crystal superalloys Rafting in single-crystal superalloys occurs as a consequence of high temperature, stress level and stress state, the two-phase nature of the material and time during which the material is subjected to load (4), (5), (6) and (7). The driving force for rafting is related to the diffusion of atoms from one phase to the 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 horizontal matrix channel width is approximately 60 nm. After a certain time and under a certain stress (state) and temperature, the microstructure has evolved to the intermediate state with partly developed rafts. The precipitates extend in a direction perpendicular to the loading axis and become flatter. They tend to take the shape of thin flat plates. Also, but less clearly visible in figure 4.1, the width of the matrix channels perpendicular to the loading axis increases, called 26

30 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 some locations and the opposite happens at other locations. In the fully rafted state, thus when two adjacent precipitates meet, the microstructure can be thought of as a lamellar structure of both phases. The time after which the rafting process is completed depends on the temperature, the stress and the lattice parameter of both phases. The lattice parameter is important as it causes a certain misfit between the phases. In CMSX-4 the misfit is called negative which means that the lattice constant of the precipitates phase is slightly smaller than the lattice constant of the matrix phase. The misfit interferes with the externally applied stress and initiates diffusion of atoms between the two phases. This is explained in 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 components on rafting in the single-crystal superalloy CMSX-4. Stress-free state (left) and loaded state (right) external stress is applied, the only stresses present are the misfit stresses found at the interfaces of both phases. The chemical potentials of both phases at point 1 and 2 (γ in the lower right of the left figure) are equal. Under these circum- 27

31 stances, due to the presence of misfit stress, rafting will occur in all direction at an equal rate (also called nondirectional coarsening). In the right-hand side of the figure, however, a tensile stress is applied. The resulting stresses at locations 1 and 2 are not equal in this case. The local stresses in both phases are modified in the direction of the external stress. In the direction perpendicular to the external stress axis stresses change due to the different Poisson numbers of the matrix and precipitate, however, this effect is small. All stress components combine to yield one specific hydrostatic stress σ h (σ h = 1 3 (σ 1 + σ 2 + σ 3 )). This hydrostatic stress is known to directly influence the chemical potentials of atoms. Tensile stresses act in the γ -phase parallel to the γ-channels. These are balanced by the compressive stresses in the γ-channels parallel to the γ - surfaces which are much larger than small tensile stress components which act perpendicular to the channels. Rafting is driven by the tendency of the system to decrease its overall γ /γinterface energy. Therefore σ h and thus the chemical potential of atoms which is directly influenced by σ h, see (10), are different. Differences in chemical potentials of atoms are known to act as driving force for diffusion (11). To summarize, the chemical potentials of the atoms at locations 1 and 2 which were equal in case of zero external stress are not longer equal in the case when the external stress is nonzero. This stress difference leads to a driving force for diffusion and results finally in the observed rafting effect. The net effect is thus the diffusion of γ -atoms from the top and bottom surface of the precipitate (thus near the interface with horizontal matrix channels) towards the vertical matrix channels, thereby filling up the space which was initially filled with matrix phase. As the precipitate atoms move from the top and bottom surface of the precipitate, this space is filled with the matrix atoms from the horizontal channels. Effectively, the width of the horizontal channels is thus increased and the width of the vertical matrix channels is decreased. 4.2 Effect of rafting on the macroscopic material behavior The effect of rafting in single-crystal superalloys has remained not well understood during a quite long period. Experimental studies showed controversial results until it was recognized that the effect of rafting could not be assessed for a general set of testing conditions but rather for specific temperature and stress ranges. One of the explanations for the increasing creep behavior as the result of rafting was based on the assumption that dislocation glide, which is one of the deformation mechanisms, is eliminated due to the lamellar structure of precipitate and matrix phase. Further deformation should occur by the mechanism of precipitate shearing, which requires stresses to be sufficiently high. For stresses below 150 MPa, precipitate shearing is most likely not to occur even at relatively high temperatures. The result is a higher resistance to deformation and increased creep behavior. On the other hand, when the temperature and stress are relatively high (> 950 and > 150 MPa, respectively), the relative escape rate of interfacial dislocations is reduced so the dislocation path is decreased and also the stress is high enough to enable precipitate shearing, deformation is more easily realized. This results in a deteriorated macroscopic creep behavior. 28